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				<a name="h1" id="h1"></a>
				<h1><a href="http://namesonnodes.org/" class="title">Names on Nodes</a>: <span class="langname">MathML</span> Definitions (Version 1.2)</h1>
				<p class="authors">T. Michael Keesey<!--<sup><a href="#aff1">*</a></sup>--></p>
				<p class="affiliations"><!--<a name="aff1" id="aff1"></a><strong>*</strong> -->P.O. Box 292304 Los Angeles, CA, USA 90027; <a href="mailto:keesey@gmail.com?subject=Names+on+Nodes%3AMathML+Definitions"><code>keesey@gmail.com</code></a></p>
				<p class="publication-date">First Draft Published Online 2009 July 29</p>
				<p class="publication-date">Version 1.0 Published Online 2010 February 21</p>
				<p class="publication-date">Version 1.2 Published Online 2010 April 2</p>
				<p class="skip">&#x2193; <a href="#section-DefinitionsMath">Skip to the definitions.</a></p>
				<div id="section-Abstract">
					<h2>Abstract</h2>
					<p>
						Phylogenetic nomenclature is a type of biological nomenclature which ties taxonomic names to taxa using algorithms that rely on phylogeny (i.e., patterns of ancestry and descent).
						Unlike earlier forms of biological nomenclature (e.g., Linnaean, rank-based), the application of a name to a taxon is unambiguous under an appropriate phylogenetic hypothesis.
						Phylogeny may be modeled as directed, acyclic graphs, with taxonomic units as vertices and immediate ancestor-descendant relations (i.e., parent-child relations) as arcs (directed edges) connecting them.
						Because phylogeny can be modeled mathematically, phylogenetic definitions may be expressed as mathematical formulae.
						Mathematical formulae, in turn, may be expressed using an extensible computer language called <span class="langname">MathML</span>.
						Expressing phylogenetic definitions in <span class="langname">MathML</span> requires the definition of certain entities.
						Here I review the relevant mathematical and biological concepts, terms, and notations, and provide an overview of <span class="langname">MathML</span>.
						I correlate these concepts to each other and, finally, define the entities needed to express phylogenetic definitions in <span class="langname">MathML</span>.
					</p>
				</div>
				<div id="section-TOC">
					<h2>Table of Contents</h2>
					<table>
						<tr>
							<th colspan="3"><a href="#section-Abstract">Abstract</a></th>
						</tr>
						<tr>
							<th colspan="3"><a href="#section-TOC">Table of Contents</a></th>
						</tr>
						<tr>
							<th rowspan="4"><a href="#section-Introduction">Introduction</a></th>
							<th colspan="2"><a href="#section-History">History</a></th>
						</tr>
						<tr>
							<th style="min-width: 16em"><a href="#section-ReviewMath">A General Review of Mathematical Concepts, Terms, and Notation</a></th>
							<td>
								<a href="#section-Collections">Collections</a> &#x2022; 
								<a href="#section-Sets">Sets</a> &#x2022;
								<a href="#section-Lists">Lists</a> &#x2022;
								<a href="#section-Relations">Relations</a> &#x2022;
								<a href="#section-Graphs">Graphs</a> &#x2022;
								<a href="#section-Functions">Functions</a>
							</td>
						</tr>
						<tr>
							<th><a href="#section-ReviewBio">A General Review of Biological Concepts and Terms</a></th>
							<td>
								<a href="#section-TaxonomicUnits">Taxonomic Units</a> &#x2022; 
								<a href="#section-Taxonomy">Taxonomy</a> &#x2022;
								<a href="#section-Specimens">Specimens</a> &#x2022;
								<a href="#section-CharacterStates">Character States</a> &#x2022;
								<a href="#section-Phylogeny">Phylogeny</a> &#x2022;
								<a href="#section-Lineages">Lineages</a> &#x2022;
								<a href="#section-Cladogens">Cladogens</a> &#x2022;
								<a href="#section-Clades">Clades</a> &#x2022;
								<a href="#section-NonClades">Non-Clades</a> &#x2022;
								<a href="#section-PhylogenyBasedNomenclature">Phylogeny-Based Nomenclature</a>
							</td>
						</tr>
						<tr>
							<th><a href="#section-ReviewMathML">A General Review of <span class="langname">MathML</span> and Its Foundational Technologies</a></th>
							<td>
								<a href="#section-Strings">Strings</a> &#x2022; 
								<a href="#section-URIs">URIs</a> &#x2022;
								<a href="#section-Namespaces">Namespaces</a> &#x2022;
								<a href="#section-XML" class="langname">XML</a> &#x2022;
								<a href="#section-MathML" class="langname">MathML</a>
							</td>
						</tr>
						<tr>
							<th colspan="2"><a href="#section-CorrelateBioMath">On the Correlation of Biological and Mathematical Terms</a></th>
							<td>
								<a href="#section-TaxaSets">Taxa and Sets</a> &#x2022; 
								<a href="#section-AncestryPrecedence">Ancestry and Precedence</a> &#x2022;
								<a href="#section-PhylogenyGraphs">Phylogeny and Graphs</a>
							</td>
						</tr>
						<tr>
							<th colspan="2" rowspan="2"><a href="#section-DefinitionsMath">Definitions of Mathematical Entities</a></th>
							<td>
								<a href="#definitions-format">Format</a> &#x2022; <a href="#definitions-context">Phylogenetic Context</a>
							</td>
						</tr>
						<tr>
							<td>
								<a href="#def-PhylogeneticGraph">Phylogenetic Graph</a> &#x2022;
								<a href="#def-UniversalTaxon">Universal Taxon</a> &#x2022; 
								<a href="#def-Maximal">Maximal</a> &#x2022;
								<a href="#def-Minimal">Minimal</a> &#x2022;
								<a href="#def-PredecessorUnion">Predecessor Union</a> &#x2022;
								<a href="#def-PredecessorIntersection">Predecessor Intersection</a> &#x2022;
								<a href="#def-SuccessorUnion">Successor Union</a> &#x2022;
								<a href="#def-SuccessorIntersection">Successor Intersection</a> &#x2022;
								<a href="#def-ExclusivePredecessors">Exclusive Predecessors</a> &#x2022;
								<a href="#def-SynapomorphicPredecessors">Synapomorphic Predecessors</a> &#x2022;
								<a href="#def-Clade">Clade</a> &#x2022;
								<a href="#def-CrownClade">Crown Clade</a> &#x2022;
								<a href="#def-TotalClade">Total Clade</a>
							</td>
						</tr>
						<tr>
							<th colspan="2"><a href="#appendix1">Appendix I</a></th>
              <td><a href="#appendix1">Implemented <span class="langname">MathML</span> Elements</a></td>
						</tr>
						<tr>
							<th colspan="2"><a href="#appendix2">Appendix II</a></th>
              <td><a href="#appendix2">Equivalence Between Biological and Mathematical Terms</a></td>
						</tr>
						<tr>
							<th colspan="3"><a href="#supplements">Supplementary Files</a></th>
						</tr>
					</table>
				</div>
				<a id="section-Introduction"></a>
				<h2>Introduction</h2>
				<a id="section-History"></a>
				<h3>History</h3>
				<p>
				  Most of the phylogenetic and nomenclatural concepts are taken from the <a href="http://phylocode.org" class="title">International Code of Phylogenetic Nomenclature</a> (or, the <span class="title">PhyloCode</span> for short) and <a href="http://www.ohio.edu/phylocode/citations.html">literature cited by it</a>. 
				  The initial discussions on creating a mathematical foundation for phylogenetic nomenclature took place circa 2001 on the <span class="title">PhyloCode</span> Mailing List, with Nathan Wilson first suggesting algorithms for node- and branch-based definitions.
				  In 2004, I began planning an application entitled <a href="http://namesonnodes.org/" class="title">Names on Nodes</a> (originally <span class="title">Names on NEXUS</span>) which would automate the application of phylogenetically-defined names to phylogenetic hypotheses, using the algorithms first proposed on the mailing list.
				  I developed these concepts further and presented them at the Second International Phylogenetic Nomenclature Meeting at Yale University, New Haven, Connecticut in 2006.
				  (See <a href="http://dx.doi.org/10.1111/j.1463-6409.2006.00268.x">the report by Laurin and Cantino</a>.)
				  In 2007, I authored a paper entitled <a href="http://dx.doi.org/10.1111/j.1463-6409.2007.00302.x" class="title">A Mathematical Approach to Defining Clade Names, with Potential Applications to Computer Storage and Processing</a>, detailing the mathematical concepts and how they might be represented in <span class="langname">MathML</span>.
				  This document represents a refinement of the concepts in my 2007 paper. Much is the same or similar. Differences include (but are not limited to):
        </p>
				<ul>
				  <li>Changes to some of the notation and terminology.</li>
				  <li>Removal of the cladogen (formerly "cladogenetic set") functions and alteration of clade functions.</li>
				  <li>Better integration with <span class="langname">MathML</span>.</li>
				  <li>Emphasis on taxonomic units, rather than individual organisms, and on contextual assertions, rather than absolute facts.</li>
				</ul>
				<p>
				  Some concepts from my 2007 paper have been omitted (notably those involving distance metrics), but may be included in a future version.
        </p>
        <p>
          <a href="http://namesonnodes.org/" class="title">Names on Nodes</a> is an open-source project.
          The progress of the project, including this document, can be viewed online at <a href="http://code.google.com/p/namesonnodes/"><code>http://code.google.com/p/namesonnodes/</code></a> (this website, older versions of the project) and <a href="http://bitbucket.org/keesey/namesonnodes-sa/"><code>http://bitbucket.org/keesey/namesonnodes-sa/</code></a> (the standalone version of the application).
        </p>
				<a id="section-ReviewMath"></a>
				<h3>A General Review of Mathematical Concepts, Terms, and Notation</h3>
				<p class="skip">&#x2193; <a href="#section-ReviewBio">Skip this section.</a></p>
				<p>
					Some terminology and notation varies across different contexts.
					Where possible, I have followed <span class="langname">MathML</span>'s terminology and default notation.
					Some exceptions have been made for certain logical symbols which are more easily read as words than as symbolic characters, e.g., <span class="math">and</span> instead of <span class="math">&and;</span>, <span class="math">for all</span> instead of <span class="math">&forall;</span>, etc.
				</p>
				<p>
					The symbol <span class="math"><strong>:=</strong></span> means "is defined as".
				</p>
				<a id="section-Collections"></a>
				<h4>Collections</h4>
				<p>
					A <strong>collection</strong> is an entity which consists of zero or more distinct objects.
					Objects in a collection are <strong>members</strong> of the collection.
					A collection with one member is a <strong>singleton</strong>.
				</p>
				<a id="section-Sets"></a>
				<h4>Sets</h4>
				<div class="figure-right">
					<img src="./images/set.png" width="200" height="124" alt=""/>
					<p>
						<span class="header">Set with members.</span>
						This diagram ilustrates the set <span class="math">{<i>x</i>, <i>y</i>, <i>z</i>}</span>.
						Elements <span class="a"><i>a</i></span> and <span class="a"><i>b</i></span> are not members.
					</p>
				</div>
				<p>
					A <strong>set</strong> is an unordered collection.
					When an object, <span class="math"><i>x</i></span>, is a member of a set, <span class="math"><i>S</i></span>, this is denoted <span class="math"><i>x</i> &isin; <i>S</i></span>.
					Sets may themselves be members of other sets.
					Sets may be denoted in the following ways:
				</p>
				<ul>
					<li>Extensionally, as a list of members: <span class="math">{<i>x</i>, <i>y</i>, <i>z</i>}</span></li>
					<li>Intensionally, with a rule that determines membership: <span class="math">{<i>x</i> | <i>x</i> &gt; 1}</span>, <span class="math">{<i>x</i> | <i>x</i> exhibits a cellular nucleus}</span></li>
					<li>Using a defined symbol or name: <span class="math">&#x2205;</span>, <span class="math">U</span>, <span class="math nomen">Mammalia</span>, <span class="math">YPM-VP 1450</span></li>
				</ul>
				<p>
					An intensional definition may explicitly limit members to a given superset: <span class="math">{<i>x</i> &isin; <i>Mammalia</i> | <i>x</i> is extant}</span>.
				</p>
				<p>
					The <strong>empty set</strong> is the set which includes no members, denoted as <span class="math">&#x2205;</span>.
				</p>
				<!--<p>
					The symbol <span class="math">&#x211D;</span> indicates the set of all <strong>real numbers</strong>.
					The nonnegative real numbers are indicated as <span class="math">&#x211D;<sub>0</sub><sup>+</sup></span>.
				</p>-->
				<!--<p>
					The <strong>cardinality</strong> of a set is the number of members in that set, denoted with enclosing vertical bars: <span class="math">|<i>S</i>|</span>.
				</p>
				<p>
					<span class="header">Examples.</span>
					<span class="math">|&#x2205;| = 0</span>.
					<span class="math">|{1, 2, 3}| = 3</span>.
				</p>-->
				<p>
					If all members of a set, <span class="math"><i>A</i></span>, are members of a set, <span class="math"><i>B</i></span>, then <span class="math"><i>A</i></span> is a <strong>subset</strong> of <span class="math"><i>B</i></span>, denoted <span class="math"><i>A</i> &#x2286; <i>B</i></span>.
					<span class="math"><i>B</i></span> is a <strong>superset</strong> of <span class="math"><i>A</i></span>.
					Note that all sets are subsets and supersets of themselves.
					If <span class="math"><i>A</i> &#x2286; <i>B</i></span> and <span class="math"><i>A</i> &#x2260; <i>B</i></span>, then <span class="math"><i>A</i></span> is a <strong>proper subset</strong> of <span class="math"><i>B</i></span>, denoted <span class="math"><i>A</i> &#x2282; <i>B</i></span>, and <span class="math"><i>B</i></span> is a <strong>proper superset</strong> of <span class="math"><i>A</i></span>.
					Note that &#x2205; is a subset of all sets, and a proper subset of all nonempty sets.
				</p>
				<div class="figure-right">
					<img src="./images/set-operations.png" width="200" height="372" alt=""/>
					<p>
						<span class="header">Set operations.</span>
						From top to bottom: union, intersection, and difference.
					</p>
				</div>
				<p>
					The operations of <strong>union</strong>, <strong>intersection</strong>, and <strong>difference</strong> may be applied to sets:
				</p>
				<ul>
					<li>
						<span class="header">Union.</span>
						<span class="math"><i>A</i> &#x222A; <i>B</i> := {<i>x</i> | <i>x</i> &isin; <i>A</i> or <i>x</i> &isin; <i>B</i>}</span>
					</li>
					<li>
						<span class="header">Intersection.</span>
						<span class="math"><i>A</i> &#x2229; <i>B</i> := {<i>x</i> | <i>x</i> &isin; <i>A</i> and <i>x</i> &isin; <i>B</i>}</span>
					</li>
					<li>
						<span class="header">Difference.</span>
						<span class="math"><i>A</i> &minus; <i>B</i> := {<i>x</i> &isin; <i>A</i> | <i>x</i> &notin; <i>B</i>}</span>
					</li>
				</ul>
				<p>
					<span class="header">Examples.</span>
					If <span class="math"><i>A</i> = {1, 2}</span> and <span class="math"><i>B</i> = {2, 3}</span>, then <span class="math"><i>A</i> &#x222A; <i>B</i> = {1, 2, 3}</span>, <span class="math"><i>A</i> &#x2229; <i>B</i> = {2}</span>, <span class="math"><i>A</i> &minus; <i>B</i> = {1}</span>, and <span class="math"><i>B</i> &minus; <i>A</i> = {3}</span>.
					Note that the order of arguments only matters for set difference.
				</p>
				<p>
					A <strong>partition</strong> of a set, <span class="math"><i>S</i></span>, is a set of subsets of <span class="math"><i>S</i></span>, such that no sets in the partition overlap and all members of <span class="math"><i>S</i></span> are members of some set in the partition.
					A partition, <span class="math"><i>P</i><sub>1</sub></span>, is a <strong>refinement</strong> of another partition, <span class="math"><i>P</i><sub>2</sub></span>, if every member of <span class="math"><i>P</i><sub>1</sub></span> is a subset of some member of <span class="math"><i>P</i><sub>2</sub></span>.
					<span class="math"><i>P</i><sub>1</sub></span> is <strong>finer</strong> than <span class="math"><i>P</i><sub>2</sub></span>, and <span class="math"><i>P</i><sub>2</sub></span> is <strong>coarser</strong> than <span class="math"><i>P</i><sub>1</sub></span>.
					This is written <span class="math"><i>P</i><sub>1</sub> &le; <i>P</i><sub>2</sub></span>.
				</p>
				<p>
					<span class="header">Example.</span>
					If <span class="math"><i>S</i> = {1, 2, 3}</span>, then the partitions of <span class="math"><i>S</i></span> are <span class="math">{&#x2205;, <i>S</i>}</span>, <span class="math">{&#x2205;, {1}, {2, 3}}</span>, <span class="math">{&#x2205;, {1, 2}, {3}}</span>, <span class="math">{&#x2205;, {1, 3}, {2}}</span>, and <span class="math">{&#x2205;, {1}, {2}, {3}}</span>.
					The partition <span class="math">{&#x2205;, {1}, {2}, {3}}</span> is a refinement of <span class="math">{&#x2205;, {1}, {2, 3}}</span>, which is a refinement of <span class="math">{&#x2205;, <i>S</i>}</span>.
				</p>
				<p>
					The <strong>power set</strong> of a set, <span class="math"><i>S</i></span>, is the set of all subsets of <span class="math"><i>S</i></span>, denoted <span class="math">2<sup><i>S</i></sup></span>.
				</p>
				<p>
					<span class="header">Example.</span>
					If <span class="math"><i>S</i> = {1, 2, 3}</span>, then <span class="math">2<sup><i>S</i></sup> = {&#x2205;, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, <i>S</i>}</span>.
				</p>
				<a id="section-Lists"></a>
				<h4>Lists</h4>
				<p>
					A <strong>list</strong> is an ordered collection of elements, denoted as a series of elements within brackets, e.g., <span class="math">[<i>x</i>, <i>y</i>, <i>z</i>]</span>.
					Unlike sets, lists may have the same member multiple times, e.g., <span class="math">[<i>x</i>, <i>x</i>, <i>y</i>]</span>.
					A list with two members is an <strong>ordered pair</strong>.
					A list with three members is an <strong>ordered triple</strong>.
					A list with <span class="math"><i>n</i></span> members is an <strong><span class="math"><i>n</i></span>-tuple</strong>.
					The <span class="math"><i>n</i></span>th member of a list, <span class="math"><i>p</i></span>, is denoted <span class="math"><i>p</i><sub><i>n</i></sub></span>.
				</p>
				<p>
					<span class="header">Example.</span>
					If <span class="math"><i>p</i> = [<i>x</i>, <i>y</i>, <i>z</i>]</span>, then <span class="math"><i>p</i><sub>1</sub> = <i>x</i></span>, <span class="math"><i>p</i><sub>2</sub> = <i>y</i></span>, and <span class="math"><i>p</i><sub>3</sub> = <i>z</i></span>.
					The list, <span class="math"><i>p</i></span>, is an ordered triple (3-tuple).
				</p>
				<p>
					A <strong>Cartesian product</strong> of two sets, <span class="math"><i>A</i></span> and <span class="math"><i>B</i></span>, is the set of all ordered pairs wherein the first element is a member of <span class="math"><i>A</i></span> and the second element is a member of <span class="math"><i>B</i></span>.
					This product is denoted as <span class="math"><i>A</i> &times; <i>B</i></span>.
					The product <span class="math"><i>A</i> &times; <i>A</i></span> may also be denoted as <span class="math"><i>A</i><sup>2</sup></span>.
					Cartesian products may be generalized to cover any whole number, <span class="math"><i>n</i></span>, of sets, in which case the members of the product are <span class="math"><i>n</i></span>-tuples.
				</p>
				<p>
					<span class="header">Example.</span>
					<span class="math">{1, 2} &times; {3, 4} = {[1, 3], [1, 4], [2, 3], [2, 4]}</span>.
				</p>
				<a id="section-Relations"></a>
				<h4>Relations</h4>
				<p>
					A <strong>relation</strong> is a set of ordered pairs.
					If <span class="math">[<i>x</i>, <i>y</i>]</span> is a pair in the relation <span class="math"><i>R</i></span>, then this is denoted as <span class="math"><i>x</i> <i>R</i> <i>y</i></span>.
					The first element in such a pair may be termed the <strong>predecessor</strong>, and the second element the <strong>successor</strong>.
					If <span class="math"><i>x</i> <i>R</i> <i>y</i></span> and <span class="math"><i>x</i> &#x2260; <i>y</i></span>, then <span class="math"><i>x</i></span> is a <strong>proper predecessor</strong> of <span class="math"><i>y</i></span> and <span class="math"><i>y</i></span> is a <strong>proper successor</strong> of <span class="math"><i>x</i></span>.
					If <span class="math"><i>x</i> <i>R</i> <i>z</i></span> and there is no other element, <span class="math"><i>y</i></span>, such that <span class="math"><i>x</i> <i>R</i> <i>y</i></span> and <span class="math"><i>y</i> <i>R</i> <i>z</i></span>, then <span class="math"><i>x</i></span> is an <strong>immediate predecessor</strong> of <span class="math"><i>z</i></span> and <span class="math"><i>z</i></span> is an <strong>immediate successor</strong> of <span class="math"><i>x</i></span>.
				</p>
				<p>
					The expression <span class="math"><i>R</i>[<i>x</i>]</span> denotes the set <span class="math">{<i>y</i> | <i>x</i> <i>R</i> <i>y</i>}</span>.
					For example, <span class="math">&gt;[0]</span> indicates the set of all negative numbers.
				</p>
				<p>
					A <strong>partial order</strong> is a relation with the properties of <strong>reflexivity</strong>, <strong>antisymmetry</strong>, and <strong>transitivity</strong>:
				</p>
				<ul>
					<li>
						<span class="header">Reflexivity.</span>
						<span class="math">For all <i>x</i>, <i>x</i> <i>R</i> <i>x</i></span>.
					</li>
					<li>
						<span class="header">Antisymmetry.</span>
						<span class="math">For all <i>x</i>, <i>y</i>, if <i>x</i> <i>R</i> <i>y</i> and <i>x</i> <i>R</i> <i>y</i>, then <i>y</i> <i>R</i> <i>x</i></span>.
					</li>
					<li>
						<span class="header">Transitivity.</span>
						<span class="math">For all <i>x</i>, <i>y</i>, <i>z</i>, if <i>x</i> <i>R</i> <i>y</i> and <i>y</i> <i>R</i> <i>z</i>, then <i>x</i> <i>R</i> <i>z</i></span>.
					</li>
				</ul>
				<p>
					The <strong>transitive closure</strong> of a relation, <span class="math"><i>R</i></span>, is the smallest (i.e., least inclusive) transitive relation that includes <span class="math"><i>R</i></span>. (If <span class="math"><i>R</i></span> is transtiive then it is its own transitive closure.)
				</p>
				<p>
					If <span class="math"><i>R</i></span> is a partial order and <span class="math"><i>x</i> <i>R</i> <i>y</i></span> or <span class="math"><i>y</i> <i>R</i> <i>x</i></span>, then <span class="math"><i>x</i></span> and <span class="math"><i>y</i></span> are <strong>comparable</strong>.
					If all elements in a set can be compared to each other, then the set is a <strong>chain</strong>.
					If no two different elements in a set are comparable, then it is an <strong>antichain</strong>.
				</p>
				<a id="section-Graphs"></a>
				<h4>Graphs</h4>
				<div class="figure-left">
				  <img src="./images/graphs.png" width="200" height="372" alt=""/>
				  <p>
				    <span class="header">Graphs.</span>
				    Circles represent vertices, lines represent edges, and arrows represent arcs (directed edges).
				    From top to bottom, an undirected graph, a directed graph (or digraph) which is cyclic, and a directed graph which is acyclic. 
          </p>
				</div>
				<p>
					A <strong>graph</strong> is an entity containing a set of objects, called <strong>vertices</strong>, and connections of the vertices, called <strong>edges</strong>.
					A graph may be defined as a type of ordered pair, in which the first element is the <strong>vertex set</strong> and the second element is the <strong>edge set</strong>: <span class="math">[<i>V</i>, <i>E</i>]</span>.
					In an <strong>undirected graph</strong>, each edge is a set of two vertices, indicating that those vertices are connected, or <strong>incident</strong>.
					In a <strong>directed graph</strong>, or <strong>digraph</strong>, each edge is an ordered pair of vertices, indicating that the first element, or <strong>head</strong>, connects to the second element, or <strong>tail</strong>.
					Edges in directed graphs may also be called <strong>arcs</strong>.
				</p>
				<p>
					A <strong>walk</strong> in a graph is a list of vertices in which each vertex is incident to the next vertex in the list.
					A <strong>path</strong> is a walk in a directed graph wherein some arc in the graph points from each vertex to the next vertex in the list.
					A <strong>cycle</strong> is a path which begins and ends with the same vertex.
					A directed graph is said to be <strong>acyclic</strong> if there are no cycles in it.
				</p>
				<a id="section-Functions"></a>
				<h4>Functions</h4>
				<p>
					A <strong>function</strong> maps an element, called an <strong>argument</strong>, to a value.
					Formally, a function may be defined as an ordered triple of three sets: <span class="math"><i>f</i> := [<i>X</i>, <i>Y</i>, <i>F</i>]</span>.
					The final set, <span class="math"><i>F</i></span>, is a set of ordered pairs, wherein the first element (a member of <span class="math"><i>X</i></span>) is the argument and the second element (a member of <span class="math"><i>Y</i></span>) is the value.
					There can be only one ordered pair per argument.
					If an ordered pair, <span class="math">[<i>x</i>, <i>y</i>]</span>, is a member of <span class="math"><i>F</i></span>, this may be denoted as <span class="math"><i>f</i>(<i>x</i>) = <i>y</i></span>.
					If the argument is a list, instead of <span class="math"><i>f</i>([<i>a</i><sub>1</sub>, <i>a</i><sub>2</sub>, &#x2026; <i>a</i><sub><i>n</i></sub>])</span>, it is customary to simply write <span class="math"><i>f</i>(<i>a</i><sub>1</sub>, <i>a</i><sub>2</sub>, &#x2026; <i>a</i><sub><i>n</i></sub>)</span>.
					Sometimes this may be written <span class="math"><i>a</i><sub>1</sub> <i>f</i> <i>a</i><sub>2</sub> <i>f</i> &#x2026; <i>a</i><sub><i>n</i></sub></span> (infix notation).
				</p>
				<p>
					The set including all of a function's arguments is the <strong>domain</strong>.
					All values of the function are within the <strong>codomain</strong>.
					The set of all values is the <strong>image</strong>, which is a subset of the codomain.
					If a function, <span class="math"><i>f</i></span>, has domain <span class="math"><i>X</i></span> and codomain <span class="math"><i>Y</i></span>, this is denoted as <span class="math"><i>f</i>: <i>X</i> &rarr; <i>Y</i></span>.
				</p>
				<p>
					The <b>composite</b> of two functions, <span class="math"><i>f</i></span> and <span class="math"><i>g</i></span>, is a function which uses the value of <span class="math"><i>g</i></span> as an argument for <span class="math"><i>f</i></span>.
					Composition is written <span class="math"><i>f</i> &#x2218; <i>g</i></span>, so that <span class="math">(<i>f</i> &#x2218; <i>g</i>)(<i>x</i>) = <i>f</i>(<i>g</i>(<i>x</i>))</span>.
					Note that the codomain of <span class="math"><i>g</i></span> must be a subset of the domain of <span class="math"><i>f</i></span>.
					If <span class="math"><i>g</i> : <i>X</i> &rarr; <i>Y</i></span> and <span class="math"><i>f</i> : <i>Y</i> &rarr; <i>Z</i></span>, then <span class="math">(<i>f</i> &#x2218; <i>g</i>) : <i>X</i> &rarr; <i>Z</i></span>.
				</p>
				<!--p>
					A <b>metric</b> on a set, <span class="math"><i>X</i></span>, is a function with <span class="math"><i>X</i><sup>2</sup></span> as its domain and the set of nonnegative real numbers as its codomain.
					It defines a <strong>metric distance</strong> between any two members of <span class="math"><i>X</i></span>.
					The <strong><span class="math"><i>&#x3B5;</i></span>-ball</strong> of <span class="math"><i>x</i></span> is the set of all elements less than a certain distance, <span class="math"><i>&#x3B5;</i></span>, from <span class="math"><i>x</i></span>.
				</p-->
				<a id="section-ReviewBio"></a>
				<h3>A General Review of Biological Concepts and Terms</h3>
				<p class="skip">&#x2193; <a href="#section-ReviewMathML">Skip this section.</a></p>
				<a id="section-TaxonomicUnits"></a>
				<h4>Taxonomic Units</h4>
				<p>
					Within the context of a study, life forms are divided into discrete sets.
					The finest sets (i.e., the sets which are not subdivided) may be referred to as <strong>taxonomic units</strong> or simply <strong>units</strong>.
					A unit may represent any of the following levels of biological organization:
				</p>
				<ul>
					<li>
						An <strong>organism</strong>, i.e., an individual living entity.
						This kind of unit may be conceptualized as a singleton set. 
					</li>
					<li>
						A <strong>population</strong>, i.e., a set of interbreeding and/or closely related organisms, generally within a given spatiotemporal region.
						Some populations are recognized as <strong>species</strong> according to various criteria. 
					</li>
					<li>
						A more inclusive set of related organisms.
					</li>
				</ul>
				<p>
					In all cases, all members of a taxonomic unit must be related to each other in ways that do not involve organisms outside the unit.
				</p>
				<p>
				  It is important to emphasize that what constitutes a taxonomic unit is context-dependent.
				  A unit in one context may be a superset of a unit in a different context.
        </p>
				<p>
					An <strong>operational unit</strong> is a unit represented by concrete data.
					A <strong>hypothetical unit</strong> is a unit whose existence is inferred.
				</p>
				<p>
					A unit may be considered <strong>extant</strong> (living as of or after a certain time) or <strong>extinct</strong> (no longer living at that time).
				</p>
				<a id="section-Taxonomy"></a>
				<h4>Taxonomy</h4>
				<p>
					A nonempty set of organisms is a <strong>taxon</strong> (plural: <strong>taxa</strong>).
					Taxonomic units are the least inclusive taxa used in a given context.
					A taxon may be conceptualized as a union of some number (one or more) of taxonomic units (which is how <a href="http://namesonnodes.org/" class="title">Names on Nodes</a> treats them).
				</p>
				<p>
					A taxon whose members are all within another taxon is a <strong>subtaxon</strong> of that other taxon.
					A taxon which includes all members of another taxon is a <strong>supertaxon</strong> of that taxon.
					The most inclusive taxon is the <strong>universal taxon</strong>, which is a superset of all taxa.
					A unit which is a subtaxon of another taxon is a <strong>subunit</strong>.
					(Note that there is no such thing as a "superunit"; if a taxon has subsets in a certain context, then it is not a unit.)
				</p>
				<p>
					A <strong>taxonomy</strong> is a scheme or system for recognizing certain taxa and relating them as supertaxa and subtaxa (or, in some cases, overlapping taxa).
				</p>
				<p>
					A taxon is extant if any of its members are extant, and extinct if all of it members are extinct.
				</p>
				<p>
					A <strong>taxonomic name</strong> is a word or series of words which signifies a taxon.
					A <strong>nomenclatural code</strong> is a set of rules which taxonomic names may be governed by.
				</p>
				<a id="section-Specimens"></a>
				<h4>Specimens</h4>
				<p>
					In addition to taxonomic names, taxa may also be referenced using <strong>specimens</strong>.
					A specimen is an object which has been catalogued as part of a <strong>specimen collection</strong>.
					A specimen collection is often indicated by an abbreviation of its name, specified within the context, e.g., "Yale Peabody Museum: Vertebrate Paleontology Collection" may be abbreviated as "YPM-VP".
					A specimen within a collection may be indicated by the collection's name or abbreviation followed by an <strong>identifier</strong> that is unique within the collection, e.g., <span class="math">YPM-VP 1450</span>.
					A specimen may have multiple identifiers if it has been transferred from one collection to another. For example, <span class="math">AMNH 973</span> and <span class="math">CM 9380</span> are the same specimen.
				</p>
				<p>
					A specimen may represent no organisms (e.g., a mineralogical specimen), one organism (e.g., a fossil skeleton), or multiple organisms (e.g., a microbe slide).
				</p>
				<a id="section-CharacterStates"></a>
				<h4>Character States</h4>
				<p>
					Taxa may be defined intensionally using a description of a necessary criterion, that is, a <strong>character state</strong>.
					Organisms exhibiting the state are members of the taxon. Valid states must be discrete and absolute, that is, organisms cannot partially exhibit them.
				</p>
				<p>
					<span class="header">Examples.</span>
					"<span class="math">Cellular nucleus present</span>" is a valid state, assuming that "cellular nucleus" has been defined in such a way that it cannot be only partially present.
					"<span class="math">Large leaf size</span>" is not a valid state, since it is relative, not absolute.
				</p>
				<p>
					Taxa may be defined using a set of character states.
					If all of the states are required for membership, the taxon is <strong>monothetic</strong>.
					If it is only required that one or more of the states be exhibited, the taxon is <strong>polythetic</strong>.
				</p>
				<a id="section-Phylogeny"></a>
				<h4>Phylogeny</h4>
				<div class="figure-right">
				  <img src="./images/phylogeny.png" width="200" height="218" alt=""/>
				  <p>
				    <span class="header">Phylogeny as a graph.</span>
				    Circles (vertices) represent organisms or taxonomic units.
				    Arrows (arcs) represent immediate descent.
				  </p>
				</div>
				<p>
					Every organism has one or more <strong>ancestors</strong> and/or one or more <strong>descendants</strong>.
					An immediate ancestor is a <strong>parent</strong>, and an immediate descendant is a <strong>child</strong>.
					The pattern of ancestry and descent among organisms is <strong>phylogeny</strong>.
				</p>
				<p>
					Taxa, including taxonomic units, may also be related in terms of ancestry and descent.
					If all members of one taxon, <span class="math"><i>A</i></span>, are ancestral to all members of another taxon, <span class="math"><i>B</i></span>, then <span class="math"><i>A</i></span> is ancestral to <span class="math"><i>B</i></span>.
				</p>
				<p>
					A <strong>phylogenetic hypothesis</strong> is an arrangement of taxonomic units into ancestor-descendant relationships.
					The <strong>resolution</strong> of a phylogenetic hypothesis refers to the size of its taxonomic units.
					For example, a hypothesis with the highest resolution would use singleton units representing individual organisms, while a hypothesis with low resolution might use large taxa as units.
				</p>
				<p>
					Phylogenetic hypotheses function as contexts wherein algorithms may be applied.
					Within such a context, immediate ancestor units may be called "parents" and immediate descendant units may be called "children", with the understanding that this is not necessarily the same as parents and children at the organismal level, depending on the resolution of the hypothesis.
				</p>
				<div class="figure-left">
				  <img src="./images/minimal-maximal.png" width="200" height="109" alt=""/>
				  <p>
				    <span class="header">Minimal and maximal units.</span>
				    At left, the minimal units are shown in black;
				    at right, the maximal units.
				  </p>
				</div>
				<p>
					Although a fuller correlation will be made further on, I note here that the biological term "ancestor" correlates to the mathematical term "proper predecessor", and the biological term "descendant" correlates to the mathematical term "proper successor".
					Therefore, we may say that a <strong>predecessor</strong> of a taxonomic unit is any ancestor of that unit, or that unit itself.
					Conversely, a <strong>successor</strong> of a unit is any descendant of that unit, or that unit itself.
					I also note that the terms "maximal" and "minimal" may be applied to units with regard to their supertaxa.
					The <strong>minimal</strong> subunits of a taxon are those which are not descended from any other subunit.
					The <strong>maximal</strong> subunits of a taxon are those which are not ancestral to any other subunit.
				</p>
				<p>
				  The predecessors of a taxon constitute the union of all units which are predecessors of all subunits of that taxon.
				  The successors of a taxon constitute the union of all units which are successors of all subunits of that taxon.
					The <strong>common predecessors</strong> of a taxon constitute the intersection of the predecessors of all subunits.
					The <strong>common successors</strong> of a taxon constitute the intersection of the successors of all subunits.
				</p>
				<div class="figure-center">
				  <img src="./images/predecessors.png" width="450" height="90" alt=""/>
				  <p>
				    <span class="header">Predecessors.</span>
				    The left image highlights two units in the phylogeny.
				    The center image shows their predecessors (including the units themselves).
				    The right image shows their common predecessors.
				  </p>
				</div>
				<div class="figure-center">
				  <img src="./images/successors.png" width="300" height="109" alt=""/>
				  <p>
				    <span class="header">Successors.</span>
				    The left image highlights two units in the phylogeny.
				    The center image shows their successors (including the units themselves).
				    The right image shows their common successors.
				  </p>
				</div>
				<div class="figure-left">
				  <img src="./images/exclusive-predecessors.png" width="200" height="183" alt=""/>
				  <p>
				    <span class="header">Exclusive Predecessors.</span>
				    The top left image indicates the internal taxon, and the top right image indicates the external taxon.
				    The middle left image shows the internal predecessors, and the middle right image shows the external predecessors.
				    The bottom image shows the exclusive predecessors.
				  </p>
				</div>
				<div class="figure-right">
				  <img src="./images/apomorphic-predecessors.png" width="200" height="328" alt=""/>
				  <p>
				    <span class="header">Apomorphic Predecessors.</span>
				    From top to bottom: the representative taxon, the apomorphic taxon, the representative taxon's predecessors, and the apomorphic predecessors.
				  </p>
				</div>
				<p>
					The <strong>exclusive predecessors</strong> of a taxon, <span class="math"><i>A</i></span>, with regard to another taxon, <span class="math"><i>Z</i></span>, constitute the common predecessors of <span class="math"><i>A</i></span> except any which are predecessors of any subunit of <span class="math"><i>Z</i></span>.
					<span class="math"><i>A</i></span> may be termed the <strong>internal taxon</strong> and <span class="math"><i>Z</i></span> may be termed the <strong>external taxon</strong>.
				</p>
				<p>
					The <strong>apomorphic predecessors</strong> of a taxon, <span class="math"><i>A</i></span>, with regard to another (generally character-based) taxon, <span class="math"><i>M</i></span>, constitute the common predecessors of <span class="math"><i>A</i></span> which are also subunits of <span class="math"><i>M</i></span>.
					<span class="math"><i>A</i></span> may be termed the <strong>representative taxon</strong> and <span class="math"><i>M</i></span> may be termed the <strong>apomorphic taxon</strong>.
				</p>
				<a id="section-Lineages"></a>
				<h4>Lineages</h4>
				<p>
					A <strong>lineage</strong> is a sequence of taxonomic units wherein each unit is preceded by one of its parents and/or followed by one of its children.
				</p>
				<p>
					The <strong>synapomorphic predecessors</strong> of a taxon, <span class="math"><i>A</i></span>, with regard to taxon <span class="math"><i>M</i></span>, constitute the union of all apomorphic predecessor units for which there is at least one lineage for every subunit of <span class="math"><i>A</i></span> satisfying the following conditions:
				</p>
				<ol>
					<li>The first unit in the lineage (i.e., the ancestor of all other units) is the apomorphic predecessor unit.</li>
					<li>The last unit in the lineage (i.e., the descendant of all other units) is the subunit of <span class="math"><i>A</i></span>.</li>
					<li>All units in the lineage are subunits of <span class="math"><i>M</i></span>.</li>
				</ol>
				<div class="figure-center">
				  <img src="./images/synapomorphic-predecessors.png" width="400" height="163" alt=""/>
				  <p>
				    <span class="header">Synapomorphic Predecessors.</span>
				    The synapomorphic predecessors, using the representative taxon and the apomorphic taxon from the previous figure.
				  </p>
				</div>
				<a id="section-Cladogens"></a>
				<h4>Cladogens</h4>
				<div class="figure-right" style="width:240px">
				  <img src="./images/cladogens.png" width="220" height="423" alt=""/>
				  <p>
				    <span class="header">Cladogens.</span>
				    From top to bottom: a specifier set (N1) and its node-based cladogen (N2); an internal taxon (B1), an external taxon (B2), and the resultant branch-based cladogen (B3); a representative taxon (S1), an apomorphic taxon (S2), and the resultant synapomorphic cladogen (S3).
				  </p>
				</div>
				<p>
					A taxon which fulfills the following requirements (in a given phylogenetic context) is here termed a <strong>cladogen</strong> (new term; previously "cladogenetic set" in <a href="http://dx.doi.org/10.1111/j.1463-6409.2007.00302.x">Keesey [2007]</a>):
				</p>
				<ol>
					<li>No subunit of a cladogen can be ancestral to any other subunit.</li>
					<li>There must be at least one unit which is a common successor of all subunits of the cladogen.</li>
				</ol>
				<p>
					All taxonomic units are cladogens, but larger cladogens may include multiple subunits.
				</p>
				<p>
					A <strong>node-based cladogen</strong> consists of the maximal common predecessors of a taxon.
				</p>
				<p>
					A <strong>branch-based cladogen</strong> consists of the minimal exclusive predecessors of an internal taxon with regard to an external taxon.
				</p>
				<p>
					An <strong>apomorphy-based cladogen</strong> consists of the minimal synapomorphic predecessors of a representative taxon with regard to an apomorphic taxon.
				</p>
				<a id="section-Clades"></a>
				<h4>Clades</h4>
				<p>
				  If a taxon is the union of a cladogen and all descendants of all of the cladogen's subunits, then it is <strong>monophyletic</strong>.
				  Monophyletic taxa are called <strong>clades</strong>.
				</p>
				<p>
					A <strong>node-based clade</strong> consists of a node-based cladogen and all descendants of all of its subunits.
					A <strong>branch-based clade</strong> consists of a branch-based cladogen and all descendants of all of its subunits.
					An <strong>apomorphy-based clade</strong> consists of an apomorphy-based cladogen and all descendants of all of its subunits.
				</p>
				<p>
					A <strong>crown clade</strong> is a type of clade wherein the minimal subunits form the node-based cladogen for some union of extant units.
					A <strong>total clade</strong> is a type of clade wherein the minimal subunits form the branch-based cladogen for some internal taxon whose subunits are all extant and some external taxon whose subunits are all extant.
					Every crown clade has a corresponding total clade where the internal taxon is the union of all extant subunits of the crown clade and the external taxon is the union of of all other extant units.
				</p>
        <div class="figure-center">
				  <img src="./images/crown-total.png" width="630" height="147" alt=""/>
				  <p>
				    <span class="header">Crown and total clades.</span>
				    If the image at left represents all extant taxonomic units, then the image in the center represents a crown clade and the image at right represents the corresponding total clade. 
				  </p>
				</div>
				<a id="section-NonClades"></a>
				<h4>Non-Clades</h4>
        <div class="figure-right">
				  <img src="./images/non-clades.png" width="200" height="405" alt=""/>
				  <p>
				    <span class="header">Non-clades.</span>
				    From top to bottom, using the same phylogeny as in the previous figure: a paraphyletic taxon, a stem taxon, and a polyphyletic taxon.  
				  </p>
				</div>
				<p>
					If a taxon's minimal subunits form a cladogen, but the taxon does not include all descendants of that cladogen, then it is <strong>paraphyletic</strong>.
					(Note that cladogens themselves are paraphyletic, with the exception of taxonomic units that have no descendants.
					Such terminal taxonomic units are clades themselves, since they consist of a cladogen and all of its descendants.)
				</p>
				<p>
				  A special type of paraphyletic taxon is a <strong>stem taxon</strong>, which is formed by subtracting a crown clade from its corresponding total clade. 
				</p>
				<p>
					If the minimal subunits of a taxon do not form a cladogen, then that taxon is <strong>polyphyletic</strong>.
				</p>
				<a id="section-PhylogenyBasedNomenclature"></a>
				<h4>Phylogeny-Based Nomenclature</h4>
				<p>
					A taxonomic name may be strictly defined under a phylogenetic hypothesis by using a <strong>phylogeny-based definition</strong>.
					Most commonly, such names refer to clades, but other types of taxa may also be phylogenetically referenced.
				</p>
				<p>
					A <strong>phylogeny-based code</strong> is a nomenclatural code which may be used to govern phylogeny-based definitions. Currently there are no such codes in effect, but there is a draft of one called the <a href="http://phylocode.org/"><i>International Code of Phylogenetic Nomenclature</i></a> (or the <i>PhyloCode</i>, for short).
					This code is intended to go into effect in the next few years, exist alongside the rank-based codes, and govern clade names across all biological disciplines.
				</p>
				<a id="section-ReviewMathML"></a>
				<h3>A General Review of <span class="langname">MathML</span> and Its Foundational Technologies</h3>
				<p class="skip">&#x2193; <a href="#section-CorrelateBioMath">Skip this section.</a></p>
				<a id="section-Strings"></a>
				<h4>Strings</h4>
					<p>
						A <strong>string</strong> is a sequence of characters.
						Strings which are meant to be interpreted by a computer are referred to as <strong>code</strong>.
						Literal strings are referred to as <strong>text</strong>.
						A string which identifies an object is a <strong>name</strong>.
					</p>
				<a id="section-URIs"></a>
				<h4>URIs</h4>
				<p>
					A <strong>Uniform Resource Identifier</strong>, or <strong>URI</strong>, is a string identifying a resource on the Internet.
				</p>
				<p>
					One of the most common types of URI is the <strong>Uniform Resource Locator</strong>, or <strong>URL</strong>, which specifies an address and a mechanism for retrieval.
					For example, a URL identifying this document is <code>http://namesonnodes.org/ns/math/2009/</code> (<code>http</code> is the retrieval mechanism, i.e., Hypertext Transfer Protocol, and <code>namesonnodes.org/ns/math/2009</code> is the address).
				</p>
				<p>
					Another type of URL is the <strong>Uniform Resource Name</strong>, or <strong>URN</strong>, which functions as a location-independent name.
					For example, a URN identifying the species <i>Homo sapiens</i> Linnaeus 1758 is <code>urn:lsid:ubio.org:namebank:109086</code>. 
				</p>
				<p>
					For more on URIs, see these official specifications:
				</p>
				<ul>
					<li><a href="http://www.ietf.org/rfc/rfc2396" class="title">Uniform Resource Identifiers (URI): Generic Syntax</a></li>
					<li><a href="http://www.ietf.org/rfc/rfc1738" class="title">Uniform Resource Locators (URL)</a></li>
					<li><a href="http://www.ietf.org/rfc/rfc2141" class="title">URN Syntax</a></li>
				</ul>
				<a id="section-Namespaces"></a>
				<h4>Namespaces</h4>
				<p>
					Generally, a <strong>namespace</strong> is a set of names, called <strong>local names</strong>, each of which has a single meaning in the context of the namespace.
					Namespaces are commonly identified using URIs, which then function as <strong>namespace identifiers</strong>.
					In some contexts, a shorter identifier may be equated with a URI.
				</p>
				<p>
					A <strong>qualified name</strong> is an expression joining a namespace identifier with a local name.
					Different computer languages have different methods of joining these.
					Common conventions are to use one or two colons ("<code>:</code>").
				</p>
				<a id="section-XML"></a>
				<h4 class="langname">XML</h4>
				<p>
					<span class="langname"><strong>Extensible Markup Language</strong></span>, or <span class="langname"><strong>XML</strong></span> for short, is a specification for creating <strong>markup languages</strong>.
					Text in <span class="langname">XML</span> may be surrounded with <strong>tags</strong>: an <strong>opening tag</strong>, of the form <code>&lt;<i>abc</i>&gt;</code>, and a <strong>closing tag</strong>, of the form <code>&lt;/<i>abc</i>&gt;</code>, where &quot;<code><i>abc</i></code>&quot; is the name of the tag.
					For example, in the <span class="langname">XML</span> expression <code>&lt;sentence&gt;Hello, world!&lt;/sentence&gt;</code>, the text "<code>Hello, world!</code>" has been marked up by <code>sentence</code> tags.
					<span class="langname">XML</span> tags may also included nested tags, for example: <code>&lt;sentence&gt;&lt;word&gt;Hello&lt;/word&gt;, &lt;word&gt;world&lt;/word&gt;!&lt;/sentence&gt;</code>.
					The entire stucture consisting of an opening tag, content, and a closing tag is an <strong>element</strong>.
					Any elements within an element's content are <strong>child elements</strong>.
					An element with no content,  an <strong>empty element</strong>, may be written as a <strong>self-closing tag</strong>: <code>&lt;<i>abc</i>/&gt;</code>.
				</p>
				<p>
					Both opening and self-closing <span class="langname">XML</span> tags may be augmented with <strong>attributes</strong>, each of which pairs a name to a value: <code>&lt;<i>tagName</i> <i>attrName</i>="<i>attrValue</i>"/&gt;</code>.
					An <span class="langname">XML</span> tag may have any number of attributes, as long as they all have different names.
				</p>
				<p>
					Tag and attribute names may be qualified names. Consider the following <span class="langname">XML</span> code:
				</p>
				<pre>&lt;html xmlns="http://www.w3.org/1999/xhtml" xmlns:m="http://www.w3.org/1998/Math/MathML"&gt;
&nbsp;&nbsp;&nbsp;&lt;head&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;title&gt;XML Namespaces Example&lt;title&gt;
&nbsp;&nbsp;&nbsp;&lt;/head&gt;
&nbsp;&nbsp;&nbsp;&lt;body&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;div&gt;This is XHTML.&lt;div&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;div&gt;The following is MathML:&lt;div&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;m:math&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;m:apply&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;m:sin/&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;m:ci&gt;x&lt;/m:ci&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;/m:apply&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;/m:math&gt;
&nbsp;&nbsp;&nbsp;&lt;/body&gt;
&lt;/html&gt;</pre>
				<p>
					In this example, the default namespace is identified by <code>http://www.w3.org/1999/xhtml</code>, which identifies the <span class="langname">XHTML</span> namespace (for hypertext).
					A namespace identifier, <code>m</code>, is synonymized with <code>http://www.w3.org/1998/Math/MathML</code>, which identifies the <span class="langname">MathML</span> namespace (for mathematical formulae).
					Therefore, if a tag or attribute's name is unqualified, then it is interpreted as an <span class="langname">XHTML</span> name.
					If a tag or attribute's name is qualified by the prefix "<code>m:</code>", then it is interpreted as a <span class="langname">MathML</span> name.
				</p>
				<p>
					For more on <span class="langname">XML</span> and <span class="langname">XML</span> namespaces, see these official specifications:
				</p>
				<ul>
					<li class="title"><a href="http://www.w3.org/TR/2008/REC-xml-20081126/">Extensible Markup Language (XML) 1.0 (Fifth Edition)</a></li>
					<li class="title"><a href="http://www.w3.org/TR/2006/REC-xml-names-20060816/">Namespaces in XML 1.0 (Second Edition)</a></li>
				</ul>
				<a id="section-MathML"></a>
				<h4 class="langname">MathML</h4>
				<p>
					<strong><span class="langname">Mathematical Markup Language</span></strong>, or <strong><span class="langname">MathML</span></strong>, is an <span class="langname">XML</span> language for expressing mathematical concepts.
					Elements in <span class="langname">MathML</span> are divided into two major groups: <span class="langname">MathML-Presentation</span>, which contains information on how to render expressions visually, and <span class="langname">MathML-Content</span>, which models mathematical entities.
					<span class="title">Name on Nodes</span> uses a relevant subset of <span class="langname">MathML-Content</span>.
				</p>
				<p>
					An important element in <span class="langname">MathML</span> is the <code>apply</code> element.
					This indicates that the first child element is to be interpreted as an <strong>operation</strong> (i.e., a function, relation, etc.), and the subsequent child elements are to be used as arguments.
				</p>
				<p>
					<span class="header">Example.</span>
					The <span class="langname">MathML</span> element <code>&lt;apply xmlns="http://www.w3.org/1998/Math/MathML"&gt;&lt;sin/&gt;&lt;cn&gt;0&lt;/cn&gt;&lt;/apply&gt;</code> indicates that the sine function (<span class="math">sin</span>) is to be applied to the number, <span class="math">0</span> (zero).
				</p>
				<p>
					Other important elements are the <code>ci</code> (content identifier) and <code>csymbol</code> (content symbol) elements, which allow the creation of custom-defined mathematical entities.
					A <code>ci</code> element uses its textual content as a unique identifier, must be defined within the <span class="langname">MathML</span> document.
					A <code>csymbol</code> element has an attribute called <code>definitionURL</code>, which contains a URL that locates the symbol's definition.
					In <a href="http://namesonnodes.org/" class="title">Names on Nodes</a>, the value of the <code>definitionURL</code> attribute may be a URL identifying a definition in this document, e.g., <a href="#def-UniversalTaxon" class="term"><code>http://namesonnodes.org/ns/math/2009#def-UniversalTaxon</code></a>.
					Both <code>ci</code> and <code>csymbol</code> elements may contain <span class="langname">MathML-Presentation</span> elements indicating how they are to be presented.
				</p>
				<p>
					<span class="header">Example.</span>
					The following <span class="langname">MathML</span> element defines the identifier <code>&lt;ci&gt;Lycopodiophyta&lt;/ci&gt;</code> as a node-based clade consisting of all successors of the maximal common predecessors of the species <span class="nomen">Lycopodium clavatum</span>, <span class="nomen">Huperzia selago</span>, <span class="nomen">Iso&euml;tes lacustris</span>, and <span class="nomen">Selaginella apoda</span>.
					(This is equivalent to <a href="http://www.phylodiversity.net/donoghue/publications/MJD_papers/2007/164_Cantino_Taxon07.pdf">Cantino et al.'s [2007]</a> definition of the clade name "<span class="nomen">Lycopodiophyta</span>".)
				</p>
				<pre>&lt;declare xmlns="http://www.w3.org/1998/Math/MathML"&gt;
&nbsp;&nbsp;&nbsp;&lt;ci&gt;Lycopodiophyta&lt;/ci&gt;
&nbsp;&nbsp;&nbsp;&lt;apply&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;csymbol definitionURL="http://namesonnodes.org/ns/math/2009#def-Clade"/&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;apply&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;union/&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;ci&gt;Lycopodium clavatum&lt;/ci&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;ci&gt;Huperzia selago&lt;/ci&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;ci&gt;Isoetes lacustris&lt;/ci&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;ci&gt;Selaginella apoda&lt;/ci&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;/apply&gt;
&nbsp;&nbsp;&nbsp;&lt;/apply&gt;
&lt;/declare&gt;</pre>
				<p>
					This assumes that the identifiers <code>&lt;ci&gt;Lycopodium clavatum&lt;/ci&gt;</code>, <code>&lt;ci&gt;Huperzia selago&lt;/ci&gt;</code>, <code>&lt;ci&gt;Isoetes lacustris&lt;/ci&gt;</code>, and <code>&lt;ci&gt;Selaginella apoda&lt;/ci&gt;</code> have been defined elsewhere.
					The entire expression might be presented visually as: <span class="math"><i>Lycopodiophyta</i> := Clade(<i>Lycopodium clavatum</i> &#x222A; <i>Huperzia selago</i> &#x222A; <i>Isoetes lacustris</i> &#x222A; <i>Selaginella apoda</i>)</span>.
				</p>
				<p>For more on <span class="langname">MathML</span>, see this official specification: <a href="http://www.w3.org/TR/MathML2/" class="title">Mathematical Markup Language (MathML) Version 2.0 (Second Edition)</a>.</p>
				<a id="section-CorrelateBioMath"></a>
				<h2>On the Correlation of Biological and Mathematical Terms</h2>
				<p class="skip">&#x2193; <a href="#section-DefinitionsMath">Skip this section.</a></p>
				<a id="section-TaxaSets"></a>
				<h3>Taxa and Sets</h3>
				<p>
					As mentioned, taxa are a type of set.
					Thus, the operations defined for sets may be employed with taxa.
					Let <span class="math">U</span> be the universal taxon, the set that includes all organisms.
					To be a taxon, a set must be a nonempty subset of <span class="math">U</span>.
					In the context of a particular hypothesis, <span class="math">U</span> may be interpreted as the union of all taxonomic units.
				</p>
				<p>
					A union of taxa, <span class="math"><i>T</i><sub>1</sub> &#x222A; <i>T</i><sub>2</sub> &#x222A; &#x2026; &#x222A; <i>T</i><sub><i>n</i></sub></span>, constitutes a polythetic taxon.
					An intersection of taxa, <span class="math"><i>T</i><sub>1</sub> &#x2229; <i>T</i><sub>2</sub> &#x2229; &#x2026; &#x2229; <i>T</i><sub><i>n</i></sub></span>, constitutes a monothetic taxon.
				</p>
				<p>
					A set of taxonomic units is a partition on some subset of <span class="math">U</span>.
					Such a set forms part of a phylogenetic hypothesis.
					The resolution of the hypothesis is determined by the fineness or coarseness of the set of taxonomic units.
				</p>
				<a id="section-AncestryPrecedence"></a>
				<h3>Ancestry and Precedence</h3>
				<p>
					Parenthood may be defined as an antisymmetric, nontransitive relation. 
					Let the relation <span class="math">&#x22B2; := {[<i>x</i>, <i>y</i>] | <i>x</i></span> is a parent of <span class="math"><i>y</i>}</span>.
					The expression <span class="math"><i>x</i> &#x22B2; <i>y</i></span> means that <span class="math"><i>x</i></span> is a parent, or immediate predecessor, of <span class="math"><i>y</i></span>.
					The inverse relation, <span class="math">&#x22B3;</span>, is childhood.
					The symmetric relation, <span class="math">&#x22C8;</span>, may be used thusly: <span class="math"><i>x</i> &#x22C8; <i>y</i></span> if and only if <span class="math"><i>x</i> &#x22B2; <i>y</i></span> or <span class="math"><i>x</i> &#x22B3; <i>y</i></span>.
					The expression <span class="math"><i>x</i> &#x22B4; <i>y</i></span> means that <span class="math"><i>x</i></span> is a parent of or equal to <span class="math"><i>y</i></span>, that is, <span class="math"><i>x</i></span> either immediately precedes or equals <span class="math"><i>y</i></span>.
					The expression <span class="math">&#x22B4;[<i>x</i>]</span> represents the set of <span class="math"><i>x</i></span> and all of its children (immediate successors).
				</p>
				<p>
					Ancestry may be defined as the transitive closure of parenthood.
					Let the relation <span class="math">&#x227A; := {[<i>x</i>, <i>y</i>] | <i>x</i></span> is an ancestor of <span class="math"><i>y</i>}</span>.
					The expression <span class="math"><i>x</i> &#x227A; <i>y</i></span> means that <span class="math"><i>x</i></span> is an ancestor, or proper predecessor, of <span class="math"><i>y</i></span>.
					The inverse relation, <span class="math">&#x227B;</span>, is descent.
					The expression <span class="math"><i>x</i> &#x227C; <i>y</i></span> means that <span class="math"><i>x</i></span> is an ancestor of or equal to <span class="math"><i>y</i></span>, that is, <span class="math"><i>x</i></span> is a predecessor of <span class="math"><i>y</i></span>.
					The expression <span class="math">&#x227C;[<i>x</i>]</span> represents the set of <span class="math"><i>x</i></span> and all of its successors.
					The expression <span class="math">&#x227D;[<i>x</i>]</span> represents the set of <span class="math"><i>x</i></span> and all of its predecessors.
				</p>
				<p>
					These relations may be used for organisms, or, in the context of a particular hypothesis, for taxonomic units.
				</p>
				<a id="section-PhylogenyGraphs"></a>
				<h3>Phylogeny and Graphs</h3>
				<p>
					A phylogenetic hypothesis may be modeled as a directed, acyclic graph (which correlates to a partially ordered set).
					Let <span class="math"><i>T</i></span> be a set whose members are all taxonomic units in the hypothesis.
					Then a <strong>phylogenetic graph</strong> may be defined as <span class="math">[<i>T</i>, &#x22B2;]</span> (with the understanding that <span class="math">&#x22B2;</span> is a context-specific version of the relation which only includes members of <span class="math"><i>T</i><sup>2</sup></span>).
					The arcs (directed edges) in the graph point from parent units to their child units, so that the head of each arc is a parent and the tail of each arc is a child.
				</p>
				<p>
					A path in a phylogenetic graph represents a lineage from ancestor to descendant.
					An <span class="math"><i>x</i>&ndash;<i>y</i></span> path in a phylogenetic graph is a sequence of vertices (taxonomic units), <span class="math"><i>p</i></span>, of length <span class="math"><i>n</i></span> such that <span class="math"><i>x</i> = <i>p</i><sub>1</sub></span> and <span class="math"><i>y</i> = <i>p</i><sub><i>n</i></sub></span> and <span class="math"><i>p</i><sub>1</sub> &#x22B2; <i>p</i><sub>2</sub> &#x22B2; &#x2026; &#x22B2; <i>p</i><sub><i>n</i></sub></span>.
				</p>
				<p>
					A cladogen is an antichain in a phylogenetic graph wherein all subunits share at least one common successor.
					As noted earlier, each vertex in the graph (i.e., each taxonomic unit) is a cladogen.
				</p>
				<p>
					Relatedness may be represented as an undirected graph.
					Let <span class="math"><i>T</i></span> be a set whose members are all taxonomic units in a phylogenetic hypothesis.
					Then a <strong>relatedness graph</strong> may be defined as <span class="math">[<i>T</i>, {{<i>X</i>, <i>Y</i>} &#x2286; <i>T</i> | <i>X</i> &#x22C8; <i>Y</i>}]</span>.
					If two vertices (taxonomic units) in this graph are connected, then they are in some way related.
					(Note that all known organisms are theorized to be related.)
				</p>
				<a id="section-DefinitionsMath"></a>
				<h2>Definitions of Mathematical Entities</h2>
				<a id="definitions-format"></a>
				<h3>Format</h3>
				<p>The following information is given for each entity defined in this document:</p>
				<ul>
					<li>
						<span class="header">Definition URL.</span>
						The full, canonical location of the definition.
						This is to be used as the <code>definitionURL</code> attribute's value in <code>csymbol</code> elements.
					</li>
					<li>
						<span class="header">Symbol.</span>
						The symbol used for the entity in <a href="/" class="title">Names on Nodes</a>.
					</li>
					<li>
						<span class="header">Class.</span>
						The general class which this entity belongs to (set, list, or function).
					</li>
					<li>
						<span class="header">Definition.</span>
						The definition (generally mathematical) of the entity.
					</li>
					<li>
						<span class="header">Discussion.</span>
						Further discussion of the entity.
					</li>
					<li>
						<span class="header">Implementation.</span>
						A qualified name identifying the ActionScript class which implements the entity in <a href="/" class="title">Names on Nodes</a>.
					</li>
					<li>
						<span class="header">Example(s).</span>
						One or more examples of how the entity may be used in <span class="langname">MathML</span> code.
						(See below, about <a href="#definitions-context">Phylogenetic Context</a>.)
						Each example includes:
						<ul>
    					<li>
    						<span class="header"><span class="langname">MathML</span>.</span>
			                <span class="langname">MathML</span> code showing how the entity may be used.
			                The namespace <code>http://www.w3.org/1998/Math/MathML</code> is implicit throughout all examples.
    					</li>
    					<li>
    						<span class="header">Formula.</span>
                The <span class="langname">MathML</span> code translated as a symbolic formula.
                (Not included in examples where the illustration includes it.)
    					</li>
						  <li>
    						<span class="header">Illustration.</span>
    						An illustration of the example.
                For functions, this includes illustrations of the arguments.
                (Not included in all examples.)
              </li>
    					<li>
    						<span class="header">Discussion.</span>
    						Further discussion of the entity.
    					</li>
						</ul>
					</li>
				</ul>
				<a id="definitions-context"></a>
				<h3>Phylogenetic Context</h3>
				<p>(The XML namespace <code>http://www.w3.org/1998/Math/MathML</code> is implicit throughout this section.)</p>
				<p>
				  All definitions assume the existence of a phylogenetic graph, <span class="math">[T, &#x22B2;]</span>, where <span class="math">T</span> is the set of all taxonomic units and <span class="math">&#x22B2;</span> is the relation of parent units (immediate predecessors) to child units (immediate successors).
          For the definition examples, let us assume the phylogenetic graph <span class="math">[T, &#x22B2;]</span> where <span class="math">T</span>'s members are:
        </p>
        <ul>
          <li>
            The extinct zoological genera <span class="nomen">Diadectes</span>, <span class="nomen">Dimorphodon</span>, <span class="nomen">Ichthyornis</span>, <span class="nomen">Pterodactylus</span>, and <span class="nomen">Tyrannosaurus</span>.
            These may be referred to using <span class="langname">MathML</span> code of the form <code>&lt;ci&gt;<i>Genus</i>&lt;/ci&gt;</code>, where <code><i>Genus</i></code> is the unit's generic name.
          </li>
          <li>
            The extinct zoological species <span class="nomen">Equus stenonis</span>.
            This may be referred to using the <span class="langname">MathML</span> code of the form <code>&lt;ci&gt;Equus stenonis&lt;/ci&gt;</code>.
          </li>
          <li>
            The extant zoological species <span class="nomen">Crocodylus niloticus</span> (Nile crocodiles), <span class="nomen">Equus africanus</span> (asses), <span class="nomen">Equus ferus</span> (horses), <span class="nomen">Homo sapiens</span> (humans), <span class="nomen">Lacerta agilis</span> (sand lizards), <span class="nomen">Ornithorhynchus anatinus</span> (platypuses), <span class="nomen">Struthio camelus</span> (ostriches), <span class="nomen">Testudo graeca</span> (spur-thighed tortoises), <span class="nomen">Tinamus major</span> (great tinamous), <span class="nomen">Vespertilio murinus</span> (parti-colored bats), and <span class="nomen">Vultur gryphus</span> (Andean condors).
            These may be referred to using <span class="langname">MathML</span> code of the form <code>&lt;ci&gt;<i>Genus species</i>&lt;/ci&gt;</code>, where <code><i>Genus species</i></code> is the unit's binomial.
          </li>
          <li>
            A singleton taxon including an extant <span class="math"><i>Equus africanus</i> &times; <i>Equus ferus</i></span> hybrid, labelled <span class="math">mule</span>.
            This may be referred to using the <span class="langname">MathML</span> code <code>&lt;ci&gt;&lt;mtext&gt;mule&lt;/mtext&gt;&lt;/ci&gt;</code>. 
          </li>
          <li>
            A singleton taxon including an extant <span class="math"><i>Equus ferus</i> &times; <i>Equus africanus</i></span> hybrid, labelled <span class="math">hinny</span>.
            This may be referred to using the <span class="langname">MathML</span> code <code>&lt;ci&gt;&lt;mtext&gt;hinny&lt;/mtext&gt;&lt;/ci&gt;</code>. 
          </li>
          <li>
            Thirteen hypothetical taxonomic units, labelled <span class="math">HTU<sub>1</sub></span>, <span class="math">HTU<sub>2</sub></span>, &#x2026; <span class="math">HTU<sub>13</sub></span>.
            These may be referred to using <span class="langname">MathML</span> code of the form <code>&lt;ci&gt;&lt;mphantom&gt;&lt;mo&gt;HTU&lt;/mo&gt;&lt;msub&gt;&lt;mn&gt;<i>n</i>&lt;mn&gt;&lt;/msub&gt;&lt;/mphantom&gt;&lt;/ci&gt;</code>, where <code><i>n</i></code> is the unit's number.
            All hypothetical taxonomic units are considered extinct.
          </li>
        </ul>
        <p>
        	The extant units are <span class="nomen">Crocodylus niloticus</span>, <span class="nomen">Equus africanus</span>, <span class="nomen">Equus ferus</span>, <span class="nomen">Homo sapiens</span>, <span class="nomen">Lacerta agilis</span>, <span class="nomen">Ornithorhynchus anatinus</span>, <span class="nomen">Struthio camelus</span>, <span class="nomen">Testudo graeca</span>, <span class="nomen">Tinamus major</span>, <span class="nomen">Vespertilio murinus</span>, <span class="nomen">Vultur gryphus</span>, <span class="math">hinny</span>, and <span class="math">mule</span>.
        	The union of these units may be referred to using the <span class="langname">MathML</span> code <code>&lt;ci&gt;&lt;ms&gt;extant&lt;/ms&gt;&lt;/ci&gt;</code>.
        </p>
        <p>
          Let <span class="math">&#x22B2;</span> be defined for <span class="math">T</span> as <span class="math">{[HTU<sub>1</sub>, <i>Diadectes</i>], [HTU<sub>1</sub>, HTU<sub>2</sub>], [HTU<sub>2</sub>, HTU<sub>3</sub>], [HTU<sub>2</sub>, HTU<sub>6</sub>], [HTU<sub>3</sub>, <i>O. anatinus</i>], [HTU<sub>3</sub>, HTU<sub>4</sub>], [HTU<sub>4</sub>, <i>H. sapiens</i>], [HTU<sub>4</sub>, HTU<sub>5</sub>], [HTU<sub>5</sub>, <i>V. murinus</i>], [HTU<sub>5</sub>, <i>E. stenonis</i>], [<i>E. stenonis</i>, <i>E. africanus</i>], [<i>E. stenonis</i>, <i>E. ferus</i>], [<i>E. africanus</i>, mule], [<i>E. africanus</i>, hinny], [<i>E. ferus</i>, mule], [<i>E. ferus</i>, hinny], [HTU<sub>6</sub>, <i>T. graeca</i>], [HTU<sub>6</sub>, <i>L. agilis</i>], [HTU<sub>6</sub>, HTU<sub>7</sub>], [HTU<sub>7</sub>, <i>C. niloticus</i>], [HTU<sub>7</sub>, HTU<sub>8</sub>], [HTU<sub>8</sub>, HTU<sub>9</sub>], [HTU<sub>8</sub>, HTU<sub>10</sub>], [HTU<sub>9</sub>, <i>Dimorphodon</i>], [HTU<sub>9</sub>, <i>Pterodactylus</i>], [HTU<sub>10</sub>, <i>Tyrannosaurus</i>], [HTU<sub>10</sub>, HTU<sub>11</sub>], [HTU<sub>11</sub>, <i>Ichthyornis</i>], [HTU<sub>11</sub>, HTU<sub>12</sub>], [HTU<sub>12</sub>, <i>V. gryphus</i>], [HTU<sub>12</sub>, HTU<sub>13</sub>], [HTU<sub>13</sub>, <i>S. camelus</i>], [HTU<sub>13</sub>, <i>T. major</i>]}</span>.
        </p>
        <div class="figure-center">
          <img src="./images/defs/phylogenetic-graph.png" width="600" height="701" alt="" class="bordered"/>
          <p>
            <span class="header">Phylogenetic graph.</span>
            An illustration of the phylogenetic graph <span class="math">[T, &#x22B2;]</span>.
            Taxonomic units that are extinct (as of 2010) are shown in the grey shaded area to the left;
            taxonomic units that are extant (as of 2010) are shown to the right.
          </p>
        </div>
        <p>
          Let us also define the apomorphic taxon <span class="math">"wings used for powered flight"</span> as <span class="math">HTU<sub>9</sub> &#x222A; HTU<sub>11</sub> &#x222A; HTU<sub>12</sub> &#x222A; HTU<sub>13</sub> &#x222A; <i>Dimorphodon</i> &#x222A; <i>Ichthyornis</i> &#x222A; <i>Pterodactylus</i> &#x222A; <i>Tinamus major</i> &#x222A; <i>Vespertilio murinus</i> &#x222A; <i>Vultur gryphus</i></span>.
          This may be referred to using the <span class="langname">MathML</span> code <code>&lt;ci&gt;&lt;ms&gt;wings used for powered flight&lt;/ms&gt;&lt;/ci&gt;</code>.
        </p>
        <div class="figure-center">
          <img src="./images/defs/apomorphic-taxon.png" width="600" height="701" class="bordered" alt=""/>
          <p>
            <span class="header">Apomorphic taxon.</span>
            The units in the apomorphic taxon <span class="math">"wings used for powered flight"</span> are highlighted in black.
            Note that the trait arose three times, in the ancestors of chiropterans (bats), pterosaurs, and avians (birds), and that the trait was lost once, in the ancestors of <span class="nomen">Struthio camelus</span> (ostriches).
          </p>
        </div>
				<a id="def-PhylogeneticGraph"></a>
				<h3>Phylogenetic Graph</h3>
				<table class="definition">
					<tr>
						<th>Definition URL</th>
						<td><code>http://namesonnodes.org/ns/math/2009#def-PhylogeneticGraph</code></td>
					</tr>
					<tr>
						<th>Symbol</th>
						<td><span class="math">P</span></td>
					</tr>
					<tr>
						<th>Class</th>
						<td>List (Directed, Acyclic Graph)</td>
					</tr>
					<tr>
						<th>Definition</th>
						<td class="math">
						  <p>T := the set of all taxonomic units.</p>
						  <p>&#x22B2; := the relation of parent units to child units.</p>
						  <p>P := [T, &#x22B2;].</p>
						</td>
					</tr>
					<tr>
						<th>Discussion</th>
						<td>
							<p>
				                <span class="math">T</span> is the set of all taxonomic units in the current phylogenetic context.
				                This is not to be confused with <span class="math">U</span>, which is the union of all taxonomic units in the current phylogenetic context.
							</p>
							<p>
								The relation <span class="math">&#x22B2;</span> includes all ordered pairs <span class="math">(<i>x</i>, <i>y</i>) &isin; T<sup>2</sup></span> such that <span class="math"><i>x</i></span> is an immediate ancestor (parent) of <span class="math"><i>y</i></span>. 
							</p>
							<p>
								<span class="math">P</span> is a directed, acyclic graph, which serves as the phylogenetic context for all other entities defined herein.
								Thus, it must be define before any entities that depend on it are used. (See Example 2, below.)
							</p>
						</td>
					</tr>
					<tr>
						<th>Implementation</th>
						<td>
							<p>
				              <span class="math">P</span> may be any implementation of <code>asmathema.collections.lists::FiniteList</code>, usually <code>asmathema.collections.lists::OrderedPair</code>.
			              </p>
							<p>
				              <span class="math">T</span> may be any implementation of <code>asmathema.collections.sets::FiniteSet</code>, usually <code>asmathema.collections.sets::HashSet</code>.
			              </p>
							<p>
				              <span class="math">&#x22B2;</span> may be any implementation of <code>asmathema.collections.sets::FiniteSet</code> and <code>asmathema.collections.operators::Operator</code>, usually <code>asmathema.collections.sets::HashSet</code>.
			              </p>
			              <p>
				              See also <code>asmathema.collections.graphs::DAGSolver</code> and <code>asmathema.collections.graphs::GraphBuilder</code>.
							</p>
			            </td>
					</tr>
				</table>
			  <div class="thead">Example 1. Selected Member</div>
				<table class="example">
				  <tbody>
  					<tr>
  						<th class="langName">MathML</th>
  						<td>
  							<pre>&lt;apply&gt;
&nbsp;&nbsp;&nbsp;&lt;selector/&gt;
&nbsp;&nbsp;&nbsp;&lt;csymbol definitionURL="http://namesonnodes.org/ns/math/2009#def-PhylogeneticGraph"/&gt;
&nbsp;&nbsp;&nbsp;&lt;cn&gt;2&lt;/cn&gt;
&lt;/apply&gt;</pre>
  						</td>
  					</tr>
  					<tr>
  						<th>Formula</th>
  						<td><span class="math">P<sub>2</sub></span></td>
  					</tr>
  					<tr>
  						<th>Discussion</th>
  						<td>
                <p>
                	This selects the second element of <span class="math">P</span>, namely, the relation <span class="math">&#x22B2;</span>.
                	The set <span class="math">T</span> may be specified in a similar manner, using <code>1</code> instead of <code>2</code>.
                </p>
              </td>
  					</tr>
  				</tbody>
				</table>
			  <div class="thead">Example 2. Definition</div>
				<table class="example">
				  <tbody>
  					<tr>
  						<th class="langName">MathML</th>
  						<td>
  							<pre>&lt;declare&gt;
&nbsp;&nbsp;&nbsp;&lt;csymbol definitionURL="http://namesonnodes.org/ns/math/2009#def-PhylogeneticGraph"/&gt;
&nbsp;&nbsp;&nbsp;&lt;list&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;set&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;ci&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;mphantom&gt;&lt;mo&gt;HTU&lt;/mo&gt;&lt;msub&gt;&lt;mn&gt;12&lt;/mn&gt;&lt;/msub&gt;&lt;/mphantom&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;/ci&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;ci&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;mphantom&gt;&lt;mo&gt;HTU&lt;/mo&gt;&lt;msub&gt;&lt;mn&gt;13&lt;/mn&gt;&lt;/msub&gt;&lt;/mphantom&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;/ci&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;ci&gt;Vultur gryphus&lt;/ci&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;ci&gt;Tinamus major&lt;/ci&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;ci&gt;Struthio camelus&lt;/ci&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;/set&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;set&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;list&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;ci&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;mphantom&gt;&lt;mo&gt;HTU&lt;/mo&gt;&lt;msub&gt;&lt;mn&gt;12&lt;/mn&gt;&lt;/msub&gt;&lt;/mphantom&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;/ci&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;ci&gt;Vultur gryphus&lt;/ci&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;/list&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;list&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;ci&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;mphantom&gt;&lt;mo&gt;HTU&lt;/mo&gt;&lt;msub&gt;&lt;mn&gt;12&lt;/mn&gt;&lt;/msub&gt;&lt;/mphantom&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;/ci&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;ci&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;mphantom&gt;&lt;mo&gt;HTU&lt;/mo&gt;&lt;msub&gt;&lt;mn&gt;13&lt;/mn&gt;&lt;/msub&gt;&lt;/mphantom&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;/ci&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;/list&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;list&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;ci&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;mphantom&gt;&lt;mo&gt;HTU&lt;/mo&gt;&lt;msub&gt;&lt;mn&gt;13&lt;/mn&gt;&lt;/msub&gt;&lt;/mphantom&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;/ci&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;ci&gt;Tinamus major&lt;/ci&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;/list&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;list&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;ci&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;mphantom&gt;&lt;mo&gt;HTU&lt;/mo&gt;&lt;msub&gt;&lt;mn&gt;13&lt;/mn&gt;&lt;/msub&gt;&lt;/mphantom&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;/ci&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;ci&gt;Struthio camelus&lt;/ci&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;/list&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;/set&gt;
&nbsp;&nbsp;&nbsp;&lt;/list&gt;
&lt;/declare&gt;</pre>
  						</td>
  					</tr>
  					<tr>
  						<th>Formula</th>
  						<td><span class="math">P := [{HTU<sub>12</sub>, HTU<sub>13</sub>, <i>Vultur gryphus</i>, <i>Tinamus major</i>, <i>Struthio camelus</i>}, {[HTU<sub>12</sub>, <i>Vultur gryphus</i>], [HTU<sub>12</sub>, HTU<sub>13</sub>], [HTU<sub>13</sub>, <i>Tinamus major</i>], [HTU<sub>13</sub>, <i>Struthio camelus</i>]}]</span>.</td>
  					</tr>
  					<tr>
  						<th>Discussion</th>
  						<td>
                <p>
                	This defines <span class="math">P</span> as a directed, acyclic graph with five vertices and four arcs.
                	This is a subgraph of the avian portion of the phylogeny used in other examples.
                </p>
              </td>
  					</tr>
  				</tbody>
				</table>
				<a id="def-UniversalTaxon"></a>
				<h3>Universal Taxon</h3>
				<table class="definition">
					<tr>
						<th>Definition URL</th>
						<td><code>http://namesonnodes.org/ns/math/2009#def-UniversalTaxon</code></td>
					</tr>
					<tr>
						<th>Symbol</th>
						<td><span class="math">U</span></td>
					</tr>
					<tr>
						<th>Class</th>
						<td>Set</td>
					</tr>
					<tr>
						<th>Definition</th>
						<td>
						  <table class="math">
						    <tr>
						      <td rowspan="2">U :=</td>
						      <td><span class="large">&#x222A;</span></td>
						      <td><i>S</i></td>
						    </tr>
						    <tr>
						      <td><span class="small"><i>S</i> &isin; T</span></td>
                </tr>
						  </table>
						</td>
					</tr>
					<tr>
						<th>Discussion</th>
						<td>
							<p>
                <span class="math">U</span> is the universal taxon, the set of all organisms.
								Operationally, <i>Names on Nodes</i> interprets <span class="math">U</span> as the union of all taxonomic units in the current phylogenetic context.
							</p>
							<p>
							 Note that this is different from <span class="math">T</span>.
							 The members of <span class="math">T</span> are taxonomic units (which are themselves sets).
							 The members of <span class="math">U</span> are organisms.
							</p>
						</td>
					</tr>
					<tr>
						<th>Implementation</th>
						<td>
							<code>org.namesonnodes.math::TaxonomicUnion</code>
						</td>
					</tr>
				</table>
				<div class="thead">Example</div>
				<table class="example">
					<tr>
						<th class="langname">MathML</th>
						<td>
							<pre>&lt;csymbol definitionURL="http://namesonnodes.org/ns/math/2009#def-UniversalTaxon"/&gt;</pre>
						</td>
					</tr>
					<tr>
					 <th>Illustration</th>
					 <td class="figure-cell">
					   <img src="./images/defs/universal-taxon.png" width="600" height="701" class="bordered" alt=""/>
					 </td>
          </tr>
					<tr>
						<th>Discussion</th>
						<td>
              <p>In the example context, <span class="math">U = <i>Crocodylus niloticus</i> &#x222A; <i>Diadectes</i> &#x222A; <i>Dimorphodon</i> &#x222A; <i>Equus africanus</i> &#x222A; <i>Equus ferus</i> &#x222A; <i>Equus stenonis</i> &#x222A; <i>Homo sapiens</i> &#x222A; <i>Ichthyornis</i> &#x222A; <i>Lacerta agilis</i> &#x222A; <i>Ornithorhynchus anatinus</i> &#x222A; <i>Pterodactylus</i> &#x222A; <i>Struthio camelus</i> &#x222A; <i>Testudo graeca</i> &#x222A; <i>Tinamus major</i> &#x222A; <i>Tyrannosaurus</i> &#x222A; <i>Vespertilio murinus</i> &#x222A; <i>Vultur gryphus</i> &#x222A; hinny &#x222A; mule &#x222A; HTU<sub>1</sub> &#x222A; HTU<sub>2</sub> &#x222A; HTU<sub>3</sub> &#x222A; HTU<sub>4</sub> &#x222A; HTU<sub>5</sub> &#x222A; HTU<sub>6</sub> &#x222A; HTU<sub>7</sub> &#x222A; HTU<sub>8</sub> &#x222A; HTU<sub>9</sub> &#x222A; HTU<sub>10</sub> &#x222A; HTU<sub>11</sub> &#x222A; HTU<sub>12</sub> &#x222A; HTU<sub>13</sub></span>.</p>
            </td>
					</tr>
				</table>
				<a id="def-Maximal"></a>
				<h3>Maximal</h3>
				<table class="definition">
					<tr>
						<th>Definition URL</th>
						<td><code>http://namesonnodes.org/ns/math/2009#def-Maximal</code></td>
					</tr>
					<tr>
						<th>Symbol</th>
						<td><span class="math">max</span></td>
					</tr>
					<tr>
						<th>Class</th>
						<td>Function</td>
					</tr>
					<tr>
						<th>Definition</th>
						<td>
							<p>
								<span class="math">max : 2<sup>U</sup> &rarr; 2<sup>U</sup></span>
								<br/>
								<span class="math">max(<i>A</i>) := {<i>x</i> &isin; <i>A</i> | for all <i>y</i> &isin; <i>A</i>, <i>x</i> &#x2280; <i>y</i>}</span>
							</p>
						</td>
					</tr>
					<tr>
						<th>Discussion</th>
						<td>
							<p>The maximal members of a taxon comprise the union of all subunits which are not ancestral to any other subunit.</p>
							<p>The concept of "maximal" correlates to what some authors have termed "last", "latest", or "most recent", as in "most recent common ancestor". However, unlike those other terms, "maximal" is not tied to chronology; the maximal members of a taxon are not necessarily contemporaries.</p>
							<p>Other potential synonyms of "maximal" are "final", "terminal", or "leafmost".</p>
							<p>The symbol for this function is the same as that of a <span class="langname">MathML</span> function, <code>max</code>; however, the <span class="langname">MathML</span> function's domain is the power set of real numbers, not the power set of the universal taxon.</p>
						</td>
					</tr>
					<tr>
						<th>Implementation</th>
						<td><code>org.namesonnodes.math.operations::Maximal</code></td>
					</tr>
				</table>
				<div class="thead">Example</div>
				<table class="example">
					<tr>
						<th class="langname">MathML</th>
						<td>
							<pre>&lt;apply&gt;
&nbsp;&nbsp;&nbsp;&lt;csymbol definitionURL="http://namesonnodes.org/ns/math/2009#def-Maximal"/&gt;
&nbsp;&nbsp;&nbsp;&lt;csymbol definitionURL="http://namesonnodes.org/ns/math/2009#def-UniversalTaxon"/&gt; 
&lt;/apply&gt;</pre>
						</td>
					</tr>
					<tr>
					 <th>Illustration</th>
					 <td class="figure-cell">
					   <img src="./images/defs/maximal.png" width="600" height="701" class="bordered" alt=""/>
					 </td>
          </tr>
          <tr>
            <th>Discussion</th>
            <td>
              <p>
                The maximal units of the universal taxon include all units with no descendants (proper successors).
                In the example context, this includes all taxonomic units except for the hypothetical taxonomic units and the species of <span class="nomen">Equus</span>.
              </p>
            </td>
          </tr>
				</table>
				<a id="def-Minimal"></a>
				<h3>Minimal</h3>
				<table class="definition">
					<tr>
						<th>Definition URL</th>
						<td><code>http://namesonnodes.org/ns/math/2009#def-Minimal</code></td>
					</tr>
					<tr>
						<th>Symbol</th>
						<td><span class="math">min</span></td>
					</tr>
					<tr>
						<th>Class</th>
						<td>Function</td>
					</tr>
					<tr>
						<th>Definition</th>
						<td>
							<p>
								<span class="math">min : 2<sup>U</sup> &rarr; 2<sup>U</sup></span>
								<br/>
								<span class="math">min(<i>A</i>) := {<i>x</i> &isin; <i>A</i> | for all <i>y</i> &isin; <i>A</i>, <i>x</i> &#x2281; <i>y</i>}</span>
							</p>
						</td>
					</tr>
					<tr>
						<th>Discussion</th>
						<td>
							<p>The minimal members of a taxon comprise the union of all subunits which are not descended from any other subunits.</p>
							<p>The concept of "minimal" correlates to what some authors have termed "earliest", "first", or "least recent", as in "least recent common ancestor". However, unlike those other terms, "minimal" is not tied to chronology; the minimal members of a taxon are not necessarily contemporaries.</p>
							<p>Other potential synonyms of "minimal" are "initial", "basalmost", or "rootmost".</p>
							<p>The symbol for this function is the same as that of a <span class="langname">MathML</span> function, <code>min</code>; however, the <span class="langname">MathML</span> function's domain is the power set of real numbers, not the power set of the universal taxon.</p>
						</td>
					</tr>
					<tr>
						<th>Implementation</th>
						<td><code>org.namesonnodes.math.operations::Minimal</code></td>
					</tr>
				</table>
				<div class="thead">Example</div>
				<table class="example">
					<tr>
						<th class="langname">MathML</th>
						<td>
							<pre>&lt;apply&gt;
&nbsp;&nbsp;&nbsp;&lt;csymbol definitionURL="http://namesonnodes.org/ns/math/2009#def-Minimal"/&gt;
&nbsp;&nbsp;&nbsp;&lt;csymbol definitionURL="http://namesonnodes.org/ns/math/2009#def-UniversalTaxon"/&gt;
&lt;/apply&gt;</pre>
						</td>
					</tr>
					<tr>
					 <th>Illustration</th>
					 <td class="figure-cell">
					   <img src="./images/defs/minimal.png" width="600" height="701" class="bordered" alt=""/>
					 </td>
          </tr>
          <tr>
            <th>Discussion</th>
            <td>
              <p>
                The maximal units of the universal taxon include all units with no ancestors (proper predecessors).
                In the example context, this is equivalent to <span class="math">HTU<sub>1</sub></span>, the only unit with no ancestors.
              </p>
            </td>
          </tr>
				</table>
				<a id="def-PredecessorUnion"></a>
				<h3>Predecessor Union</h3>
				<table class="definition">
					<tr>
						<th>Definition URL</th>
						<td><code>http://namesonnodes.org/ns/math/2009#def-PredecessorUnion</code></td>
					</tr>
					<tr>
						<th>Symbol</th>
						<td><span class="math">prc<sub>&#x222A;</sub></span></td>
					</tr>
					<tr>
						<th>Class</th>
						<td>Function</td>
					</tr>
					<tr>
						<th>Definition</th>
						<td>
							<p>
								<span class="math">prc<sub>&#x222A;</sub> : 2<sup>U</sup> &rarr; 2<sup>U</sup></span>
							</p>
							<p>
								<span class="math">prc<sub>&#x222A;</sub>(<i>A</i>) := {<i>x</i> &isin; U | for some <i>y</i> &isin; <i>A</i>, <i>x</i> &#x227c; <i>y</i>}</span>
							</p>
							or
							<table class="math">
								<tr>
									<td rowspan="2">prc<sub>&#x222A;</sub>(<i>A</i>) :=</td>
									<td><span class="large">&#x222A;</span></td>
									<td>&#x227d;[<i>x</i>]</td>
								</tr>
								<tr>
									<td><span class="small"><i>x</i> &isin; <i>A</i></span></td>
								</tr>
							</table>
						</td>
					</tr>
					<tr>
						<th>Discussion</th>
						<td>
							<p>The predecessor union of a taxon includes all members of that taxon as well as all ancestors of all members of that taxon.</p>
							<p>The predecessor union of a taxon is always a superset of the predecessor intersection.</p>
						</td>
					</tr>
					<tr>
						<th>Implementation</th>
						<td><code>org.namesonnodes.math.operations::PredecessorUnion</code></td>
					</tr>
				</table>
				<div class="thead">Example</div>
				<table class="example">
					<tr>
						<th class="langname">MathML</th>
						<td>
							<pre>&lt;apply&gt;
&nbsp;&nbsp;&nbsp;&lt;csymbol definitionURL="http://namesonnodes.org/ns/math/2009#def-PredecessorUnion"/&gt;
&nbsp;&nbsp;&nbsp;&lt;apply&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;union/&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;ci&gt;Pterodactylus&lt;/ci&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;ci&gt;Vultur gryphus&lt;/ci&gt;
&nbsp;&nbsp;&nbsp;&lt;/apply&gt;
&lt;/apply&gt;</pre>
						</td>
					</tr>
					<tr>
					 <th>Illustration</th>
					 <td class="figure-cell">
					   <img src="./images/defs/predecessor-union.png" width="600" height="701" class="bordered" alt=""/>
					 </td>
          </tr>
					<tr>
					 <th>Discussion</th>
					 <td>
					    <p>
					       The predecessor union of <span class="nomen">Pterodactylus</span> and <span class="nomen">Vultur gryphus</span> (Andean condors) includes both of those units as well as every unit ancestral to either one of them, or to both.
              </p>
					 </td>
          </tr>
				</table>
				<a id="def-PredecessorIntersection"></a>
				<h3>Predecessor Intersection</h3>
				<table class="definition">
					<tr>
						<th>Definition URL</th>
						<td><code>http://namesonnodes.org/ns/math/2009#def-PredecessorIntersection</code></td>
					</tr>
					<tr>
						<th>Symbol</th>
						<td><span class="math">prc<sub>&#x2229;</sub></span></td>
					</tr>
					<tr>
						<th>Class</th>
						<td>Function</td>
					</tr>
					<tr>
						<th>Definition</th>
						<td>
							<p>
								<span class="math">prc<sub>&#x2229;</sub> : 2<sup>U</sup> &rarr; 2<sup>U</sup></span>
							</p>
							<p>
								<span class="math">prc<sub>&#x2229;</sub>(<i>A</i>) := {<i>x</i> &isin; U | for all <i>y</i> &isin; <i>A</i>, <i>x</i> &#x227c; <i>y</i>}</span>
							</p>
							or
							<table class="math">
								<tr>
									<td rowspan="2">prc<sub>&#x2229;</sub>(<i>A</i>) :=</td>
									<td><span class="large">&#x2229;</span></td>
									<td>&#x227d;[<i>x</i>]</td>
								</tr>
								<tr>
									<td><span class="small"><i>x</i> &isin; <i>A</i></span></td>
								</tr>
							</table>
						</td>
					</tr>
					<tr>
						<th>Discussion</th>
						<td>
							<p>The predecessor intersection of a taxon includes all predecessors shared by all subunits of that taxon.</p>
							<p>The predecessor intersection of a taxon is always a subset of the predecessor union.</p>
						</td>
					</tr>
					<tr>
						<th>Implementation</th>
						<td><code>org.namesonnodes.math.operations::PredecessorIntersection</code></td>
					</tr>
				</table>
				<div class="thead">Example</div>
				<table class="example">
					<tr>
						<th class="langname">MathML</th>
						<td>
							<pre>&lt;apply&gt;
&nbsp;&nbsp;&nbsp;&lt;csymbol definitionURL="http://namesonnodes.org/ns/math/2009#def-PredecessorIntersection"/&gt;
&nbsp;&nbsp;&nbsp;&lt;apply&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;union/&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;ci&gt;Pterodactylus&lt;/ci&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;ci&gt;Vultur gryphus&lt;/ci&gt;
&nbsp;&nbsp;&nbsp;&lt;/apply&gt;
&lt;/apply&gt;</pre>
						</td>
					</tr>
					<tr>
					 <th>Illustration</th>
					 <td class="figure-cell">
					   <img src="./images/defs/predecessor-intersection.png" width="600" height="701" class="bordered" alt=""/>
					 </td>
          </tr>
					<tr>
					 <th>Discussion</th>
					 <td>
					   The predecessor intersection of <span class="nomen">Pterodactylus</span> and <span class="nomen">Vultur gryphus</span> (Andean condors) includes only those units which precede both of them.
					   This is equivalent to the ancestral lineage of <span class="nomen">Ornithodira</span>, the clade that includes <span class="nomen">Pterodactylus</span> and <span class="nomen">Vultur gryphus</span>.
					 </td>
          </tr>
				</table>
				<a id="def-SuccessorUnion"></a>
				<h3>Successor Union</h3>
				<table class="definition">
					<tr>
						<th>Definition URL</th>
						<td><code>http://namesonnodes.org/ns/math/2009#def-SuccessorUnion</code></td>
					</tr>
					<tr>
						<th>Symbol</th>
						<td><span class="math">suc<sub>&#x222A;</sub></span></td>
					</tr>
					<tr>
						<th>Class</th>
						<td>Function</td>
					</tr>
					<tr>
						<th>Definition</th>
						<td>
							<p>
								<span class="math">suc<sub>&#x222A;</sub> : 2<sup>U</sup> &rarr; 2<sup>U</sup></span>
							</p>
							<p>
								<span class="math">suc<sub>&#x222A;</sub>(<i>A</i>) := {<i>x</i> &isin; U | for some <i>y</i> &isin; <i>A</i>, <i>x</i> &#x227d; <i>y</i>}</span>
							</p>
							or
							<table class="math">
								<tr>
									<td rowspan="2">suc<sub>&#x222A;</sub>(<i>A</i>) :=</td>
									<td><span class="large">&#x222A;</span></td>
									<td>&#x227c;[<i>x</i>]</td>
								</tr>
								<tr>
									<td><span class="small"><i>x</i> &isin; <i>A</i></span></td>
								</tr>
							</table>
						</td>
					</tr>
					<tr>
						<th>Discussion</th>
						<td>
							<p>The successor union of a taxon includes all subunits of that taxon as well as all descendants of all subunits of that taxon.</p>
							<p>The successor union of a taxon is always a superset of the successor intersection.</p>
						</td>
					</tr>
					<tr>
						<th>Implementation</th>
						<td><code>org.namesonnodes.math.operations::SuccessorUnion</code></td>
					</tr>
				</table>
				<div class="thead">Example</div>
				<table class="example">
					<tr>
						<th class="langname">MathML</th>
						<td>
							<pre>&lt;apply&gt;
&nbsp;&nbsp;&nbsp;&lt;csymbol definitionURL="http://namesonnodes.org/ns/math/2009#def-SuccessorUnion"/&gt;
&nbsp;&nbsp;&nbsp;&lt;apply&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;union/&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;ci&gt;Equus stenonis&lt;/ci&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;ci&gt;Equus africanus&lt;/ci&gt;
&nbsp;&nbsp;&nbsp;&lt;/apply&gt;
&lt;/apply&gt;</pre>
						</td>
					</tr>
					<tr>
					 <th>Illustration</th>
					 <td class="figure-cell">
					   <img src="./images/defs/successor-union.png" width="600" height="701" class="bordered" alt=""/>
					 </td>
          </tr>
					<tr>
					 <th>Discussion</th>
					 <td>
					    <p>
					       The successor union of <span class="nomen">Equus stenonis</span> and <span class="nomen">Equus africanus</span> (asses) includes both of those units as well as every unit descended from either one of them, or from both.
					       Since <span class="nomen">Equus africanus</span> is descended from <span class="nomen">Equus stenonis</span>, this is the same as <span class="math"><nobr>&#x227C;[<i>Equus stenonis</i>]</nobr></span>.
              </p>
					 </td>
          </tr>
				</table>
				<a id="def-SuccessorIntersection"></a>
				<h3>Successor Intersection</h3>
				<table class="definition">
					<tr>
						<th>Definition URL</th>
						<td><code>http://namesonnodes.org/ns/math/2009#def-SuccessorIntersection</code></td>
					</tr>
					<tr>
						<th>Symbol</th>
						<td><span class="math">suc<sub>&#x2229;</sub></span></td>
					</tr>
					<tr>
						<th>Class</th>
						<td>Function</td>
					</tr>
					<tr>
						<th>Definition</th>
						<td>
							<p>
								<span class="math">suc<sub>&#x2229;</sub> : 2<sup>U</sup> &rarr; 2<sup>U</sup></span>
							</p>
							<p>
								<span class="math">suc<sub>&#x2229;</sub>(<i>A</i>) := {<i>x</i> &isin; U | for all <i>y</i> &isin; <i>A</i>, <i>x</i> &#x227d; <i>y</i>}</span>
							</p>
							or
							<table class="math">
								<tr>
									<td rowspan="2">suc<sub>&#x2229;</sub>(<i>A</i>) :=</td>
									<td><span class="large">&#x2229;</span></td>
									<td>&#x227c;[<i>x</i>]</td>
								</tr>
								<tr>
									<td><span class="small"><i>x</i> &isin; <i>A</i></span></td>
								</tr>
							</table>
						</td>
					</tr>
					<tr>
						<th>Discussion</th>
						<td>
							<p>The successor intersection of a taxon includes all all shared successors of all subunits of that taxon.</p>
							<p>The successor intersection of a taxon is always a subset of the successor union.</p>
						</td>
					</tr>
					<tr>
						<th>Implementation</th>
						<td><code>org.namesonnodes.math.operations::SuccessorIntersection</code></td>
					</tr>
				</table>
				<div class="thead">Example</div>
				<table class="example">
					<tr>
						<th class="langname">MathML</th>
						<td>
							<pre>&lt;apply&gt;
&nbsp;&nbsp;&nbsp;&lt;csymbol definitionURL="http://namesonnodes.org/ns/math/2009#def-SuccessorIntersection"/&gt;
&nbsp;&nbsp;&nbsp;&lt;apply&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;union/&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;ci&gt;Equus stenonis&lt;/ci&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;ci&gt;Equus africanus&lt;/ci&gt;
&nbsp;&nbsp;&nbsp;&lt;/apply&gt;
&lt;/apply&gt;</pre>
						</td>
					</tr>
					<tr>
					 <th>Illustration</th>
					 <td class="figure-cell">
					   <img src="./images/defs/successor-intersection.png" width="600" height="701" alt=""/>
					 </td>
          </tr>
					<tr>
					 <th>Discussion</th>
					 <td>
					    <p>
					       The successor intersection of <span class="nomen">Equus stenonis</span> and <span class="nomen">Equus africanus</span> (asses) includes only the units that succeed both of them.
					       Since <span class="nomen">Equus africanus</span> is descended from <span class="nomen">Equus stenonis</span>, this is the same as <span class="math"><nobr>&#x227C;[<i>Equus africanus</i>]</nobr></span>.
              </p>
					 </td>
          </tr>
				</table>
				<a id="def-ExclusivePredecessors"></a>
				<h3>Exclusive Predecessors</h3>
				<table class="definition">
					<tr>
						<th>Definition URL</th>
						<td><code>http://namesonnodes.org/ns/math/2009#def-ExclusivePredecessors</code></td>
					</tr>
					<tr>
						<th>Symbol</th>
						<td class="math">&larr;</td>
					</tr>
					<tr>
						<th>Class</th>
						<td>Function</td>
					</tr>
					<tr>
						<th>Definition</th>
						<td class="math">
							<p>&larr; : 2<sup>U</sup> &times; 2<sup>U</sup> &rarr; 2<sup>U</sup></p>
							<p><i>A</i> &larr; <i>Z</i> := prc<sub>&#x2229;</sub>(<i>A</i>) &minus; prc<sub>&#x222A;</sub>(<i>Z</i>)</p>
						</td>
					</tr>
					<tr>
						<th>Discussion</th>
						<td>
							<p>
								The exclusive predecessors of an internal taxon are all of its common predecessors minus the predecessor union of the external taxon.
							</p>
							<p>
								If <span class="math"><i>A</i></span> has no common predecessors, or all of those common predecessors are also predecessors of <span class="math"><i>Z</i></span>, then <span class="math"><i>A</i> &larr; <i>Z</i> = &#x2205;</span>.
							</p>
						</td>
					</tr>
					<tr>
						<th>Implementation</th>
						<td><code>org.namesonnodes.math.operations::ExclusivePredecessors</code></td>
					</tr>
				</table>
				<div class="thead">Example</div>
				<table class="example">
					<tr>
						<th class="langname">MathML</th>
						<td>
							<pre>&lt;apply&gt;
&nbsp;&nbsp;&nbsp;&lt;csymbol definitionURL="http://namesonnodes.org/ns/math/2009#def-ExclusivePredecessors"/&gt;
&nbsp;&nbsp;&nbsp;&lt;ci&gt;Vultur gryphus&lt;/ci&gt;
&nbsp;&nbsp;&nbsp;&lt;ci&gt;Pterodactylus&lt;/ci&gt;
&lt;/apply&gt;</pre>
						</td>
					</tr>
					<tr>
					 <th>Illustration</th>
					 <td class="figure-cell">
					   <img src="./images/defs/exclusive-predecessors.png" width="600" height="701" class="bordered" alt=""/>
					 </td>
          </tr>
					<tr>
					 <th>Discussion</th>
					 <td>
					   <p>
                The predecessor intersection of <span class="nomen">Vultur gryphus</span> (Andean condors) exclusive of the predecessor union of <span class="nomen">Pterodactylus</span> forms a lineage.
                The minimal unit of this lineage forms the cladogen for a branch-based clade (<span class="nomen">Dinosauromorpha</span>).
             </p>
					 </td>
          </tr>
				</table>
				<a id="def-SynapomorphicPredecessors"></a>
				<h3>Synapomorphic Predecessors</h3>
				<table class="definition">
					<tr>
						<th>Definition URL</th>
						<td><code>http://namesonnodes.org/ns/math/2009#def-SynapomorphicPredecessors</code></td>
					</tr>
					<tr>
						<th>Symbol</th>
						<td><span class="math">@</span></td>
					</tr>
					<tr>
						<th>Class</th>
						<td>Function</td>
					</tr>
					<tr>
						<th>Definition</th>
						<td>
							<p>
								<span class="math">@ : 2<sup>U</sup> &times; 2<sup>U</sup> &rarr; 2<sup>U</sup></span>
							</p>
							<p>
								<span class="math">Let P(<i>x</i>, <i>y</i>) := the set of all <i>x</i>&ndash;<i>y</i> paths in [T, &#x22B2;].</span>
							</p>
							<p>
								<span class="math">Let V(<i>p</i>) := the set of all vertices in (i.e., members of) the path <i>p</i>.</span>
							</p>
							<p>
								<span class="math"><i>M</i> @ <i>A</i> := {<i>x</i> &isin; prc<sub>&#x2229;</sub>(<i>A</i>) | for all <i>y</i> &isin; <i>A</i>, there exists some <i>p</i> &isin; P(<i>x</i>, <i>y</i>) such that V(<i>p</i>) &#x2286; <i>M</i>}</span>
							</p>
						</td>
					</tr>
					<tr>
						<th>Discussion</th>
						<td>
							<p>
								Specifying synapomorphic predecessors requires two sets, one apomorphic (<span class="math"><i>M</i></span>) and the other representative (<span class="math"><i>A</i></span>).
							</p>
							<p>
								If <span class="math"><i>A</i> &#x2288; <i>M</i></span>, then <span class="math"><i>M</i> @ <i>A</i> = &#x2205;</span>.
								There are also no synapomorphic predecessors if at least two members of <span class="math"><i>A</i></span> are in <span class="math"><i>M</i></span> due to convergence.
							</p>
						</td>
					</tr>
					<tr>
						<th>Implementation</th>
						<td><code>org.namesonnodes.math.operations::SynapomorphicPredecessors</code></td>
					</tr>
				</table>
				<div class="thead">Example</div>
				<table class="example">
					<tr>
						<th class="langname">MathML</th>
						<td>
							<pre>&lt;apply&gt;
&nbsp;&nbsp;&nbsp;&lt;csymbol definitionURL="http://namesonnodes.org/ns/math/2009#def-SynapomorphicPredecessors"/&gt;
&nbsp;&nbsp;&nbsp;&lt;ci&gt;&lt;ms&gt;wings used for powered flight&lt;/ms&gt;&lt;/ci&gt;
&nbsp;&nbsp;&nbsp;&lt;ci&gt;Vultur gryphus&lt;/ci&gt;
&lt;/apply&gt;</pre>
						</td>
					</tr>
					<tr>
					 <th>Illustration</th>
					 <td class="figure-cell">
					   <img src="./images/defs/synapomorphic-predecessors.png" width="600" height="701" class="bordered" alt=""/>
					 </td>
          </tr>
					<tr>
					 <th>Discussion</th>
					 <td>
					   <p>
					     The predecessors of <span class="nomen">Vultur gryphus</span> which share wings used for powered flight synapomorphically with <span class="nomen">Vultur gryphus</span> form a lineage of flying organisms.
               Note that this does not include any organisms with acquired flight independently.
               The minimal unit of this lineage (<span class="math">HTU<sub>11</sub></span>) forms the cladogen of an apomorphy-based clade (<span class="nomen">Avialae</span>).  
             </p>
					 </td>
          </tr>
				</table>
				<a id="def-Clade"></a>
				<h3>Clade</h3>
				<table class="definition">
					<tr>
						<th>Definition URL</th>
						<td><code>http://namesonnodes.org/ns/math/2009#def-Clade</code></td>
					</tr>
					<tr>
						<th>Symbol</th>
						<td class="math">Clade</td>
					</tr>
					<tr>
						<th>Class</th>
						<td>Function</td>
					</tr>
					<tr>
						<th>Definition</th>
						<td>
							<p class="math">Clade : 2<sup>U</sup> &rarr; 2<sup>U</sup></p>
							<table class="math">
							 <tr>
							   <td rowspan="2">
							     Clade(<i>A</i>) :=
							   </td>
							   <td rowspan="2" class="large">
							     {
							   </td>
							   <td class="piece">
							     (suc<sub>&#x222A;</sub> &#x2218; max &#x2218; prc<sub>&#x2229;</sub>)(<i>A</i>), if (suc<sub>&#x2229;</sub> &#x2218; min)(<i>A</i>) = &#x2205;.
							   </td>
							 </tr>
							 <tr>
							   <td class="piece">
							     suc<sub>&#x222A;</sub>(<i>A</i>), otherwise.
							   </td>
							 </tr>
							</table>
						</td>
					</tr>
					<tr>
						<th>Discussion</th>
						<td>
							<p>
							   This function maps any taxon to a clade, or to the empty set.
							   If the argument's minimal members do not form a cladogen, then <span class="math">Clade</span> yields the successor union of the maximal common predecessors of the argument (i.e., a node-based clade).
							   If they do form a cladogen, then <span class="math">Clade</span> yields the successor union of the argument.
							</p>
							<p>
								If (and only if) <span class="math"><i>A</i></span> has no common predecessors, then <span class="math">Clade(<i>A</i>) = &#x2205;</span>.
								Since all known organisms are theorized to descend from common ancestors, clades exist for all known taxa, in theory.
								Note that <span class="math">Clade(&#x2205;) = &#x2205;</span>.
							</p>
							<p>
								Note that, for every clade <span class="math"><i>K</i></span>, <span class="math">Clade(<i>K</i>) = <i>K</i></span>.
							</p>
						</td>
					</tr>
					<tr>
						<th>Implementation</th>
						<td><code>org.namesonnodes.math.operations::Clade</code></td>
					</tr>
				</table>
				<div class="thead">Example 1. Ancestor-Based Clade</div>
				<table class="example">
					<tr>
						<th class="langname">MathML</th>
						<td>
							<pre>&lt;apply&gt;
&nbsp;&nbsp;&nbsp;&lt;csymbol definitionURL="http://namesonnodes.org/ns/math/2009#def-Clade"/&gt;
&nbsp;&nbsp;&nbsp;&lt;ci&gt;Equus ferus&lt;/ci&gt;
&lt;/apply&gt;</pre>
						</td>
					</tr>
					<tr>
					 <th>Illustration</th>
					 <td class="figure-cell">
					   <img src="./images/defs/ancestor-based-clade.png" width="600" height="701" class="bordered" alt=""/>
					 </td>
          </tr>
          <tr>
            <th>Discussion</th>
            <td>
              <p>
                This evaluates to all successors of <span class="nomen">Equus ferus</span> (horses), which includes <span class="math">hinny</span> and <span class="math">mule</span>.
                This is an ancestor-based clade, that is, the argument (specifier) is a cladogen.
                This type of definition is not common in practice, but possible in theory.
              </p>
            </td>
          </tr>
				</table>
				<div class="thead">Example 2. Node-Based Clade</div>
				<table class="example">
					<tr>
						<th class="langname">MathML</th>
						<td>
							<pre>&lt;apply&gt;
&nbsp;&nbsp;&nbsp;&lt;csymbol definitionURL="http://namesonnodes.org/ns/math/2009#def-Clade"/&gt;
&nbsp;&nbsp;&nbsp;&lt;apply&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;union&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;ci&gt;Homo sapiens&lt;/ci&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;ci&gt;Ornithorhynchus anatinus&lt;/ci&gt;
&nbsp;&nbsp;&nbsp;&lt;/apply&gt;
&lt;/apply&gt;</pre>
						</td>
					</tr>
					<tr>
					 <th>Illustration</th>
					 <td class="figure-cell">
					   <img src="./images/defs/node-based-clade.png" width="600" height="701" class="bordered" alt=""/>
					 </td>
          </tr>
          <tr>
            <th>Discussion</th>
            <td>
              <p>
                This evaluates to all successors of the maximal common predecessors of <span class="nomen">Homo sapiens</span> (humans) and <span class="nomen">Ornithorhynchus anatinus</span> (platypuses).
                This is a node-based clade known as <span class="nomen">Mammalia</span>.
              </p>
            </td>
          </tr>
				</table>
				<div class="thead">Example 3. Node-Based Clade With Large Cladogen</div>
				<table class="example">
					<tr>
						<th class="langname">MathML</th>
						<td>
							<pre>&lt;apply&gt;
&nbsp;&nbsp;&nbsp;&lt;csymbol definitionURL="http://namesonnodes.org/ns/math/2009#def-Clade"/&gt;
&nbsp;&nbsp;&nbsp;&lt;apply&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;union&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;ci&gt;&lt;mtext&gt;mule&lt;/mtext&gt;&lt;/ci&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;ci&gt;&lt;mtext&gt;hinny&lt;/mtext&gt;&lt;/ci&gt;
&nbsp;&nbsp;&nbsp;&lt;/apply&gt;
&lt;/apply&gt;</pre>
						</td>
					</tr>
					<tr>
					 <th>Illustration</th>
					 <td class="figure-cell">
					   <img src="./images/defs/node-based-clade-large-cladogen.png" width="600" height="701" class="bordered" alt=""/>
					 </td>
          </tr>
          <tr>
            <th>Discussion</th>
            <td>
              <p>
                This evaluates to all successors of the maximal common predecessors of <span class="math">mule</span> and <span class="math">hinny</span>.
                This is a node-based clade whose cladogen, in the example context, is <span class="nomen">Equus africanus</span> &#x222A; <span class="nomen">Equus ferus</span>.
                Note that, although the cladogen consists of two units, it is still a cladogen in the example context.
                Neither unit is ancestral to the other, and they both share at least one common successor (two, in fact).
              </p>
            </td>
          </tr>
				</table>
				<div class="thead">Example 4. Branch-Based Clade</div>
				<table class="example">
					<tr>
						<th class="langname">MathML</th>
						<td>
							<pre>&lt;apply&gt;
&nbsp;&nbsp;&nbsp;&lt;csymbol definitionURL="http://namesonnodes.org/ns/math/2009#def-Clade"/&gt;
&nbsp;&nbsp;&nbsp;&lt;apply&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;csymbol definitionURL="http://namesonnodes.org/ns/math/2009#def-ExclusivePredecessors"/&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;ci&gt;Vultur gryphus&lt;/ci&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;ci&gt;Pterodactylus&lt;/ci&gt;
&nbsp;&nbsp;&nbsp;&lt;/apply&gt;
&lt;/apply&gt;</pre>
						</td>
					</tr>
					<tr>
					 <th>Illustration</th>
					 <td class="figure-cell">
					   <img src="./images/defs/branch-based-clade.png" width="600" height="701" class="bordered" alt=""/>
					 </td>
          </tr>
          <tr>
            <th>Discussion</th>
            <td>
              <p>
                This evaluates to a clade including <span class="nomen">Vultur gryphus</span> (Andean condors) and excluding <span class="nomen">Pterodactylus</span>).
                This is a branch-based clade known as <span class="nomen">Dinosauromorpha</span>.
                (Note, though, that the actual definition of <span class="nomen">Dinosauromorpha</span> is different.)
              </p>
            </td>
          </tr>
				</table>
				<div class="thead">Example 5. Apomorphy-Based Clade</div>
				<table class="example">
					<tr>
						<th class="langname">MathML</th>
						<td>
							<pre>&lt;apply&gt;
&nbsp;&nbsp;&nbsp;&lt;csymbol definitionURL="http://namesonnodes.org/ns/math/2009#def-Clade"/&gt;
&nbsp;&nbsp;&nbsp;&lt;apply&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;csymbol definitionURL="http://namesonnodes.org/ns/math/2009#def-SynapomorphicPredecessors"/&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;ci&gt;&lt;ms&gt;wings used for powered flight&lt;/ms&gt;&lt;/ci&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;ci&gt;Vultur gryphus&lt;/ci&gt;
&nbsp;&nbsp;&nbsp;&lt;/apply&gt;
&lt;/apply&gt;</pre>
						</td>
					</tr>
					<tr>
					 <th>Illustration</th>
					 <td class="figure-cell">
					   <img src="./images/defs/apomorphy-based-clade.png" width="600" height="701" class="bordered" alt=""/>
					 </td>
          </tr>
          <tr>
            <th>Discussion</th>
            <td>
              <p>
                This evaluates to the a clade including all organisms to posses wings used for powered flight synapomorphic with those of <span class="nomen">Vultur gryphus</span> (Andean condors).
                It also includes organisms whose ancestor possessed the synapomorphy, even if they themselves have lost it, e.g., <span class="nomen">Struthio camelus</span> (ostriches).
                This is an apomorphy-based clade known as <span class="nomen">Avialae</span>.
              </p>
            </td>
          </tr>
				</table>
				<a id="def-CrownClade"></a>
				<h3>Crown Clade</h3>
				<table class="definition">
					<tr>
						<th>Definition URL</th>
						<td><code>http://namesonnodes.org/ns/math/2009#def-CrownClade</code></td>
					</tr>
					<tr>
						<th>Symbol</th>
						<td class="math">Crown</td>
					</tr>
					<tr>
						<th>Class</th>
						<td>Function</td>
					</tr>
					<tr>
						<th>Definition</th>
						<td>
							<p class="math">Crown : 2<sup>U</sup> &times; 2<sup>U</sup> &rarr; 2<sup>U</sup></p>
							<p class="math">Crown(<i>A</i>, <i>E</i>) := Clade(Clade(<i>A</i>) &#x2229; <i>E</i>)</p>
						</td>
					</tr>
					<tr>
						<th>Discussion</th>
						<td>
							<p>
								Specifying a crown clade requires two taxa.
                One (<span class="math"><i>A</i></span>) is used to specify a bounding clade.
                The other (<span class="math"><i>E</i></span>) is used to indicate which units are to be considered extant.
								The crown clade is the node-based clade specified by the bounding clade's taxon's extant subunits.
							</p>
							<p>
								The bounding clade is always a superset of the crown clade.
                This includes the possibility that the clades may be identical.
								If <span class="math">Clade(<i>A</i>)</span> itself is a crown clade (using <span class="math"><i>E</i></span> as the criterion for being considered extant), then this function evaluates as <span class="math"><i>A</i></span>. 
							</p>
						</td>
					</tr>
					<tr>
						<th>Implementation</th>
						<td><code>org.namesonnodes.math.operations::CrownClade</code></td>
					</tr>
				</table>
				<div class="thead">Example 1. Node-Modified Crown Clade</div>
				<table class="example">
					<tr>
						<th class="langname">MathML</th>
						<td>
							<pre>&lt;apply&gt;
&nbsp;&nbsp;&nbsp;&lt;csymbol definitionURL="http://namesonnodes.org/ns/math/2009#def-CrownClade"&gt;
&nbsp;&nbsp;&nbsp;&lt;apply&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;union/&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;ci&gt;Tyrannosaurus&lt;/ci&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;ci&gt;Vultur gryphus&lt;/ci&gt;
&nbsp;&nbsp;&nbsp;&lt;/apply&gt;
&nbsp;&nbsp;&nbsp;&lt;ci&gt;&lt;ms&gt;extant&lt;/ms&gt;&lt;/ci&gt;
&lt;/apply&gt;</pre>
						</td>
					</tr>
					<tr>
					 <th>Illustration</th>
					 <td class="figure-cell">
					   <img src="./images/defs/node-modified-crown-clade.png" width="600" height="701" class="bordered" alt=""/>
					 </td>
          </tr>
					<tr>
					 <th>Discussion</th>
					 <td>
					   This is a node-modified crown clade definition.
					   It specifies a node-based bounding clade: <span class="math">Clade(<i>Tyrannosaurus</i> &#x222A; <i>Vultur gryphus</i>)</span>, which has the name <span class="nomen">Tyrannoraptora</span>.
					   The extant subunits of <span class="nomen">Tyrannoraptora</span> are used to specify the crown clade.
					   The end result is the crown clade <span class="nomen">Aves</span>.
					 </td>
          </tr>
				</table>
				<div class="thead">Example 2. Branch-Modified Crown Clade</div>
				<table class="example">
					<tr>
						<th class="langname">MathML</th>
						<td>
							<pre>&lt;apply&gt;
&nbsp;&nbsp;&nbsp;&lt;csymbol definitionURL="http://namesonnodes.org/ns/math/2009#def-CrownClade"&gt;
&nbsp;&nbsp;&nbsp;&lt;apply&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;csymbol definitionURL="http://namesonnodes.org/ns/math/2009#def-ExclusivePredecessors/&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;ci&gt;Vultur gryphus&lt;/ci&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;ci&gt;Pterodactylus&lt;/ci&gt;
&nbsp;&nbsp;&nbsp;&lt;/apply&gt;
&nbsp;&nbsp;&nbsp;&lt;ci&gt;&lt;ms&gt;extant&lt;/ms&gt;&lt;/ci&gt;
&lt;/apply&gt;</pre>
						</td>
					</tr>
					<tr>
					 <th>Illustration</th>
					 <td class="figure-cell">
					   <img src="./images/defs/branch-modified-crown-clade.png" width="600" height="701" class="bordered" alt=""/>
					 </td>
          </tr>
					<tr>
					 <th>Discussion</th>
					 <td>
					   This is a branch-modified crown clade definition.
					   It specifies a branch-based bounding clade: <span class="math">Clade(<i>Vultur gryphus</i> &larr; <i>Pterodactylus</i>)</span>, which has the name <span class="nomen">Dinosauromorpha</span>.
					   The extant subunits of <span class="nomen">Dinosauromorpha</span> are used to specify the crown clade.
					   The end result is the crown clade <span class="nomen">Aves</span>.
					 </td>
          </tr>
				</table>
				<div class="thead">Example 3. Apomorphy-Modified Crown Clade</div>
				<table class="example">
					<tr>
						<th class="langname">MathML</th>
						<td>
							<pre>&lt;apply&gt;
&nbsp;&nbsp;&nbsp;&lt;csymbol definitionURL="http://namesonnodes.org/ns/math/2009#def-CrownClade"&gt;
&nbsp;&nbsp;&nbsp;&lt;apply&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;csymbol definitionURL="http://namesonnodes.org/ns/math/2009#def-SynapomorphicPredecessors/&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;ci&gt;&lt;ms&gt;wings used for powered flight&lt;/ms&gt;&lt;/ci&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;ci&gt;Vultur gryphus&lt;/ci&gt;
&nbsp;&nbsp;&nbsp;&lt;/apply&gt;
&nbsp;&nbsp;&nbsp;&lt;ci&gt;&lt;ms&gt;extant&lt;/ms&gt;&lt;/ci&gt;
&lt;/apply&gt;</pre>
						</td>
					</tr>
					<tr>
					 <th>Illustration</th>
					 <td class="figure-cell">
					   <img src="./images/defs/apomorphy-modified-crown-clade.png" width="600" height="701" class="bordered" alt=""/>
					 </td>
          </tr>
					<tr>
					 <th>Discussion</th>
					 <td>
					   This is an apomorphy-modified crown clade definition.
					   It specifies an apomorphy-based bounding clade: <span class="math">Clade("wings used for powered flight" in <i>Vultur gryphus</i>)</span>, which has the name <span class="nomen">Avialae</span>.
					   The extant subunits of <span class="nomen">Avialae</span> are used to specify the crown clade.
					   The end result is the crown clade <span class="nomen">Aves</span>.
					 </td>
          </tr>
				</table>
				<a id="def-TotalClade"></a>
				<h3>Total Clade</h3>
				<table class="definition">
					<tr>
						<th>Definition URL</th>
						<td><code>http://namesonnodes.org/ns/math/2009#def-TotalClade</code></td>
					</tr>
					<tr>
						<th>Symbol</th>
						<td class="math">Total</td>
					</tr>
					<tr>
						<th>Class</th>
						<td>Function</td>
					</tr>
					<tr>
						<th>Definition</th>
						<td>
							<p class="math">Total : 2<sup>U</sup> &times; 2<sup>U</sup> &rarr; 2<sup>U</sup></p>
							<p class="math">Let <i>C</i> := Crown(<i>A</i>, <i>E</i>).</p>
							<p class="math">Total(<i>A</i>, <i>E</i>) := Clade(<i>C</i> &larr; (<i>E</i> &minus; <i>C</i>)).</p>
						</td>
					</tr>
					<tr>
						<th>Implementation</th>
						<td><code>org.namesonnodes.math.operations::TotalClade</code></td>
					</tr>
					<tr>
						<th>Discussion</th>
						<td>
							<p>
								Specifying a total clade requires two taxa.
                One (<span class="math"><i>A</i></span>) specifies the corresponding (internal) crown clade (<span class="math"><i>C</i></span>).
                The other (<span class="math"><i>E</i></span>) is used to indicate which units are to be considered extant.
								The total clade is a branch-based clade consisting of everything sharing more ancestry with the crown clade (<span class="math"><i>C</i></span>) than with anything else extant.
							</p>
						</td>
					</tr>
				</table>
				<div class="thead">Example</div>
				<table class="example">
					<tr>
						<th class="langname">MathML</th>
						<td>
							<pre>&lt;apply&gt;
&nbsp;&nbsp;&nbsp;&lt;csymbol definitionURL="http://namesonnodes.org/ns/math/2009#def-TotalClade"/&gt;
&nbsp;&nbsp;&nbsp;&lt;apply&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;union/&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;ci&gt;Struthio camelus&lt;/ci&gt;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;ci&gt;Vultur gryphus&lt;/ci&gt;
&nbsp;&nbsp;&nbsp;&lt;/apply&gt;
&nbsp;&nbsp;&nbsp;&lt;csymbol definitionURL="urn:iso:8601::2010"/&gt;
&lt;/apply&gt;</pre>
						</td>
					</tr>
					<tr>
					 <th>Illustration</th>
					 <td class="figure-cell">
					   <img src="./images/defs/total-clade.png" width="600" height="701" class="bordered" alt=""/>
					 </td>
          </tr>
					<tr>
					 <th>Discussion</th>
					 <td>
					   <p>
					     <span class="math">Crown(<i>Struthio camelus</i> &#x222A; <i>Vultur gryphus</i>, "extant") = <i>Aves</i></span>.
               <span class="math">Total(<i>Aves</i>, "extant")</span> is a branch-based clade where avians are the internal specifiers and extant non-avians are the external specifiers.
               Since, in the example context, <span class="nomen">Crocodylus niloticus</span> is the closest extant outgroup to <span class="nomen">Aves</span>, this is equivalent to <span class="math">Clade(<i>Aves</i> &larr; <i>C. niloticus</i>)</span>.
               This total clade is known by several names, including <span class="nomen">Pan-Aves</span> and <span class="nomen">Avemetatarsalia</span>. 
					   </p>
					 </td>
          </tr>
				</table>
				<a id="appendix1"></a>
				<h2>Appendix I.&mdash;Implemented <span class="langname">MathML</span> Elements</h2>
				<p>
					<span class="langname">MathML-Content</span> provides methods for modelling a wide variety of mathematical entities.
					Since <a href="http://namesonnodes.org/" class="title">Names on Nodes</a> only deals with logic and set theory, only certain elements have been implemented.
					A subset of <span class="langname">MathML-Presentation</span> has also been implemented allowing for custom rendering of identifier (<code>ci</code>) and symbol (<code>csymbol</code>) elements.
					The following is a list of all <span class="langname">MathML</span> elements which have been implemented in <a href="http://namesonnodes.org/" class="title">Names on Nodes</a>, with notes as necessary.
				</p>
				<table>
					<tr>
						<th colspan="2"><span class="langname">MathML</span> (general)</th>
					</tr>
					<tr>
						<th>
							<code>math</code>
						</th>
						<td>
						</td>
					</tr>
					<tr>
						<th colspan="2"><span class="langname">MathML-Content</span></th>
					</tr>
					<tr>
						<th>
							<code>and</code>
						</th>
						<td>
						</td>
					</tr>
					<tr>
						<th>
							<code>apply</code>
						</th>
						<td>
								The <code>type</code> attribute may be set to <code>&quot;boolean&quot;</code>, <code>&quot;fn&quot;</code>, <code>&quot;list&quot;</code>, or <code>&quot;set&quot;</code>.
						</td>
					</tr>
					<tr>
						<th>
							<code>ci</code>
						</th>
						<td>
							<p>
								The <code>type</code> attribute may be set to <code>&quot;boolean&quot;</code>, <code>&quot;fn&quot;</code>, <code>&quot;list&quot;</code>, or <code>&quot;set&quot;</code>.
							</p> 
						</td>
					</tr>
					<tr>
						<th>
							<code>cn</code>
						</th>
						<td>
							<p>
								The <code>type</code> attribute may be set to <code>&quot;integer&quot;</code>.
								Since the only use for numbers in this version of <span class="title">Names on Nodes</span> is for <code>selector</code> arguments, only integers are necessary.
							</p> 
						</td>
					</tr>
					<tr>
						<th>
							<code>compose</code>
						</th>
						<td>
						</td>
					</tr>
					<tr>
						<th>
							<code>csymbol</code>
						</th>
						<td>
							<p>
								The <code>definitionURL</code> attribute must be set to one of the following:
							</p>
							<ul>
								<li>A URL identified in the <a href="#section-DefinitionsMath" class="title">Definitions of Mathematical Entities</a> section of this document.</li>
								<li>
									A URI indicating a taxon or taxonomic name.
									Any URI not identified in this document is interpreted as a taxon.
								</li>
							</ul>
							<p>
								The <code>type</code> attribute may be set to <code>&quot;fn&quot;</code>, <code>&quot;list&quot;</code>, or <code>&quot;set&quot;</code>.
							</p> 
						</td>
					</tr>
					<tr>
						<th>
							<code>declare</code>
						</th>
						<td>
							<p>
								The <code>type</code> attribute may be set to <code>&quot;boolean&quot;</code>, <code>&quot;fn&quot;</code>, <code>&quot;list&quot;</code>, or <code>&quot;set&quot;</code>.
							</p> 
						</td>
					</tr>
					<tr>
						<th>
							<code>emptyset</code>
						</th>
						<td>
							<p>
								May be used wherever taxa may be used.
							</p>
						</td>
					</tr>
					<tr>
						<th>
							<code>eq</code>
						</th>
						<td>
						</td>
					</tr>
					<tr>
						<th>
							<code>false</code>
						</th>
						<td>
						</td>
					</tr>
					<tr>
						<th>
							<code>implies</code>
						</th>
						<td>
						</td>
					</tr>
					<tr>
						<th>
							<code>in</code>
						</th>
						<td>
						</td>
					</tr>
					<tr>
						<th>
							<code>intersect</code>
						</th>
						<td>
						</td>
					</tr>
					<tr>
						<th>
							<code>list</code>
						</th>
						<td>
						</td>
					</tr>
					<tr>
						<th>
							<code>neq</code>
						</th>
						<td>
						</td>
					</tr>
					<tr>
						<th>
							<code>not</code>
						</th>
						<td>
						</td>
					</tr>
					<tr>
						<th>
							<code>notin</code>
						</th>
						<td>
						</td>
					</tr>
					<tr>
						<th>
							<code>notprsubset</code>
						</th>
						<td>
						</td>
					</tr>
					<tr>
						<th>
							<code>notsubset</code>
						</th>
						<td>
						</td>
					</tr>
					<tr>
						<th>
							<code>or</code>
						</th>
						<td>
						</td>
					</tr>
					<tr>
						<th>
							<code>otherwise</code>
						</th>
						<td>
						</td>
					</tr>
					<tr>
						<th>
							<code>piece</code>
						</th>
						<td>
						</td>
					</tr>
					<tr>
						<th>
							<code>piecewise</code>
						</th>
						<td>
							<p>
								The <code>type</code> attribute may be set to <code>&quot;boolean&quot;</code>, <code>&quot;fn&quot;</code>, <code>&quot;list&quot;</code>, or <code>&quot;set&quot;</code>.
							</p> 
						</td>
					</tr>
					<tr>
						<th>
							<code>prsubset</code>
						</th>
						<td>
						</td>
					</tr>
					<tr>
						<th>
							<code>selector</code>
						</th>
						<td>
						</td>
					</tr>
					<tr>
						<th>
							<code>sep</code>
						</th>
						<td>
						</td>
					</tr>
					<tr>
						<th>
							<code>set</code>
						</th>
						<td>
						</td>
					</tr>
					<tr>
						<th>
							<code>setdiff</code>
						</th>
						<td>
						</td>
					</tr>
					<tr>
						<th>
							<code>subset</code>
						</th>
						<td>
						</td>
					</tr>
					<tr>
						<th>
							<code>true</code>
						</th>
						<td>
						</td>
					</tr>
					<tr>
						<th>
							<code>union</code>
						</th>
						<td>
						</td>
					</tr>
					<tr>
						<th>
							<code>xor</code>
						</th>
						<td>
						</td>
					</tr>
					<tr>
						<th colspan="2"><span class="langname">MathML-Presentation</span></th>
					</tr>
					<tr>
						<th>
							<code>merror</code>
						</th>
						<td>
						</td>
					</tr>
					<tr>
						<th>
							<code>mfenced</code>
						</th>
						<td>
						</td>
					</tr>
					<tr>
						<th>
							<code>mi</code>
						</th>
						<td>
						</td>
					</tr>
					<tr>
						<th>
							<code>mn</code>
						</th>
						<td>
						</td>
					</tr>
					<tr>
						<th>
							<code>mo</code>
						</th>
						<td>
						</td>
					</tr>
					<tr>
						<th>
							<code>mover</code>
						</th>
						<td>
						</td>
					</tr>
					<tr>
						<th>
							<code>mphantom</code>
						</th>
						<td>
							<code>mphantom</code> elements may be used to indicate that the label of a taxonomic unit should not appear in charts (e.g., for hypothetical units).
						</td>
					</tr>
					<tr>
						<th>
							<code>mrow</code>
						</th>
						<td>
						</td>
					</tr>
					<tr>
						<th>
							<code>ms</code>
						</th>
						<td>
						</td>
					</tr>
					<tr>
						<th>
							<code>mspace</code>
						</th>
						<td>
						</td>
					</tr>
					<tr>
						<th>
							<code>mstyle</code>
						</th>
						<td>
						</td>
					</tr>
					<tr>
						<th>
							<code>msub</code>
						</th>
						<td>
						</td>
					</tr>
					<tr>
						<th>
							<code>msubsup</code>
						</th>
						<td>
						</td>
					</tr>
					<tr>
						<th>
							<code>msup</code>
						</th>
						<td>
						</td>
					</tr>
					<tr>
						<th>
							<code>mtable</code>
						</th>
						<td>
						</td>
					</tr>
					<tr>
						<th>
							<code>mtd</code>
						</th>
						<td>
						</td>
					</tr>
					<tr>
						<th>
							<code>mtext</code>
						</th>
						<td>
						</td>
					</tr>
					<tr>
						<th>
							<code>mtr</code>
						</th>
						<td>
						</td>
					</tr>
					<tr>
						<th>
							<code>munder</code>
						</th>
						<td>
						</td>
					</tr>
					<tr>
						<th>
							<code>munderover</code>
						</th>
						<td>
						</td>
					</tr>
				</table>
				<a id="appendix2"></a>
				<h2>Appendix II.&mdash;Equivalence Between Biological and Mathematical Terms</h2>
				<table>
				  <tr>
				    <th>Biological Term</th>
				    <th>Mathematical Term</th>
				    <th>Notes</th>
				  </tr>
				  <tr>
				    <td>organism<br/>or<br/>individual</td>
				    <td>element<br/>or<br/>vertex</td>
				  </tr>
				  <tr>
				    <td>taxon</td>
				    <td>set</td>
				    <td>A taxon is a set of organisms.</td>
				  </tr>
				  <tr>
				    <td>subtaxon</td>
				    <td>subset</td>
				  </tr>
				  <tr>
				    <td>supertaxon</td>
				    <td>superset</td>
				  </tr>
				  <tr>
				    <td>taxonomic unit</td>
				    <td>finest set</td>
				  </tr>
				  <tr>
				    <td>phylogenetic hypothesis</td>
				    <td>directed, acyclic graph<br/>or<br/>partially-ordered set</td>
				  </tr>
				  <tr>
				    <td>ancestor</td>
				    <td>proper predecessor</td>
				  </tr>
				  <tr>
				    <td>descendant</td>
				    <td>proper successor</td>
				  </tr>
				  <tr>
				    <td>parent</td>
				    <td>immediate predecessor<br/>or<br/>head</td>
				  </tr>
				  <tr>
				    <td>child</td>
				    <td>immediate successor<br/>or<br/>tail</td>
				  </tr>
				  <tr>
				    <td>parent-child relation</td>
				    <td>directed edge<br/>or<br/>arc</td>
				  </tr>
				  <tr>
				    <td>lineage</td>
				    <td>path</td>
				    <td>A path is a type of list. The members of a path constitue a chain.</td>
				  </tr>
				  <tr>
				    <td>cladogen</td>
				    <td>antichain with a non-empty successor intersection</td>
				  </tr>
				  <tr>
				    <td>common ancestors</td>
				    <td>predecessor intersection</td>
				  </tr>
				  <tr>
				    <td>common descendants</td>
				    <td>successor intersection</td>
				  </tr>
				  <tr>
				    <td>basalmost<br/>or<br/>initial<br/>or<br/>rootmost</td>
				    <td>minimal</td>
				  </tr>
				  <tr>
				    <td>final<br/>or<br/>leafmost<br/>or<br/>terminal</td>
				    <td>maximal</td>
				  </tr>
				  <tr>
				    <td>polythetic taxon</td>
				    <td>union</td>
				  </tr>
				  <tr>
				    <td>monothetic taxon</td>
				    <td>intersection</td>
				  </tr>
				  <tr>
				    <td>related</td>
				    <td>connected</td>
 				  </tr>
				  <tr>
				    <td>filially related</td>
				    <td>comparable</td>
 				  </tr>
				</table>
				<a id="supplements"></a>
				<h2>Supplementary Files</h2>
				<table>
					<tr>
						<th>
							<a href="./mathml.xml"><code>mathml.xml</code></a>
						</th>
						<td>
							<p>
								This <span class="langname">MathML</span> file contains the context used in the definition examples of this document, along with a number of further examples.
								It may be opened with <span class="title">Names on Nodes: Standalone Version</span>.
							</p>
						</td>
					</tr>
				</table>
			</div>
		</div>
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